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m | page66-28 | 13.45 | 13.45 | A mass of $0.25 \mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\mathrm{m} / \mathrm{s}$.
If the mass is to attain a velocity of no more than $10 \mathrm{~m} / \mathrm{s}$, find the maximum height from which it can be dropped. | diff |
||
$ | page 61-10 | 89,034.79 | 89,034.79 | A home buyer can afford to spend no more than $\$ 800$ /month on mortgage payments. Suppose that the interest rate is $9 \%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.
Determine the maximum amount that this buyer can afford to borrow. | diff |
||
slugs | page216-11 | 4 | 4 | A spring is stretched 6 in by a mass that weighs $8 \mathrm{lb}$. The mass is attached to a dashpot mechanism that has a damping constant of $0.25 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}$ and is acted on by an external force of $4 \cos 2 t \mathrm{lb}$.
If the given mass is replaced by a mass $m$, determine the value of $m$ for which the amplitude of the steady state response is maximum. | diff |
||
months | page61-11 | 135.36 | 135.36 | A recent college graduate borrows $\$ 100,000$ at an interest rate of $9 \%$ to purchase a condominium. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of $800(1+t / 120)$, where $t$ is the number of months since the loan was made.
Assuming that this payment schedule can be maintained, when will the loan be fully paid? | diff |
||
Page 40 29 | 10.065778 | 10.065778 | Consider the initial value problem
$$
y^{\prime}+\frac{1}{4} y=3+2 \cos 2 t, \quad y(0)=0
$$
Determine the value of $t$ for which the solution first intersects the line $y=12$. | diff |
|||
$ | page 131-8 | 2283.63 | 2283.63 | An investor deposits $1000 in an account paying interest at a rate of 8% compounded monthly, and also makes additional deposits of \$25 per month. Find the balance in the account after 3 years. | diff |
||
m/s | page 66-28 | 11.58 | 11.58 | A mass of $0.25 \mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\mathrm{m} / \mathrm{s}$.
If the mass is dropped from a height of $30 \mathrm{~m}$, find its velocity when it hits the ground. | diff |
||
s | page202-6 | $\pi/14$ | 0.2244 | A mass of $100 \mathrm{~g}$ stretches a spring $5 \mathrm{~cm}$. If the mass is set in motion from its equilibrium position with a downward velocity of $10 \mathrm{~cm} / \mathrm{s}$, and if there is no damping, determine when does the mass first return to its equilibrium position. | diff |
||
s | page 60-6 | 130.41 | 130.41 | Suppose that a tank containing a certain liquid has an outlet near the bottom. Let $h(t)$ be the height of the liquid surface above the outlet at time $t$. Torricelli's principle states that the outflow velocity $v$ at the outlet is equal to the velocity of a particle falling freely (with no drag) from the height $h$.
Consider a water tank in the form of a right circular cylinder that is $3 \mathrm{~m}$ high above the outlet. The radius of the tank is $1 \mathrm{~m}$ and the radius of the circular outlet is $0.1 \mathrm{~m}$. If the tank is initially full of water, determine how long it takes to drain the tank down to the level of the outlet. | diff |
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page144-21 | −2 | −2 | Solve the initial value problem $y^{\prime \prime}-y^{\prime}-2 y=0, y(0)=\alpha, y^{\prime}(0)=2$. Then find $\alpha$ so that the solution approaches zero as $t \rightarrow \infty$. | diff |
|||
page156-35 | 4.946 | 4.946 | If $y_1$ and $y_2$ are a fundamental set of solutions of $t^2 y^{\prime \prime}-2 y^{\prime}+(3+t) y=0$ and if $W\left(y_1, y_2\right)(2)=3$, find the value of $W\left(y_1, y_2\right)(4)$. | diff |
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Year | Page 17 14 | 672.4 | 672.4 | Radium-226 has a half-life of 1620 years. Find the time period during which a given amount of this material is reduced by one-quarter. | diff |
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lb | Page 60-3 | 7.42 | 7.42 | A tank originally contains $100 \mathrm{gal}$ of fresh water. Then water containing $\frac{1}{2} \mathrm{lb}$ of salt per gallon is poured into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, and the mixture is allowed to leave at the same rate. After $10 \mathrm{~min}$ the process is stopped, and fresh water is poured into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional $10 \mathrm{~min}$. | diff |
||
$ | page 60-8 | 3930 | 3930 | A young person with no initial capital invests $k$ dollars per year at an annual rate of return $r$. Assume that investments are made continuously and that the return is compounded continuously.
If $r=7.5 \%$, determine $k$ so that $\$ 1$ million will be available for retirement in 40 years. | diff |
||
page164-26 | 1.8763 | 1.8763 | Consider the initial value problem
$$
y^{\prime \prime}+2 a y^{\prime}+\left(a^2+1\right) y=0, \quad y(0)=1, \quad y^{\prime}(0)=0 .
$$
For $a=1$ find the smallest $T$ such that $|y(t)|<0.1$ for $t>T$. | diff |
|||
page344-14 | 2.3613 | 2.3613 | Consider the initial value problem
$$
y^{\prime \prime}+\gamma y^{\prime}+y=\delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0,
$$
where $\gamma$ is the damping coefficient (or resistance).
Find the time $t_1$ at which the solution attains its maximum value. | diff |
|||
Page 40 28 | −1.642876 | −1.642876 | Consider the initial value problem
$$
y^{\prime}+\frac{2}{3} y=1-\frac{1}{2} t, \quad y(0)=y_0 .
$$
Find the value of $y_0$ for which the solution touches, but does not cross, the $t$-axis. | diff |
|||
$\text{day}^{-1}$ | Section 1.2, page 15 12. (a) | 0.02828 | 0.02828 | A radioactive material, such as the isotope thorium-234, disintegrates at a rate proportional to the amount currently present. If $Q(t)$ is the amount present at time $t$, then $d Q / d t=-r Q$, where $r>0$ is the decay rate. If $100 \mathrm{mg}$ of thorium-234 decays to $82.04 \mathrm{mg}$ in 1 week, determine the decay rate $r$. | diff |
||
min | page62-16 | 6.07 | 6.07 | Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of $200^{\circ} \mathrm{F}$ when freshly poured, and $1 \mathrm{~min}$ later has cooled to $190^{\circ} \mathrm{F}$ in a room at $70^{\circ} \mathrm{F}$, determine when the coffee reaches a temperature of $150^{\circ} \mathrm{F}$. | diff |
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page144-22 | -1 | -1 | Solve the initial value problem $4 y^{\prime \prime}-y=0, y(0)=2, y^{\prime}(0)=\beta$. Then find $\beta$ so that the solution approaches zero as $t \rightarrow \infty$. | diff |
|||
page145-26 | 16.3923 | 16.3923 | Consider the initial value problem (see Example 5)
$$
y^{\prime \prime}+5 y^{\prime}+6 y=0, \quad y(0)=2, \quad y^{\prime}(0)=\beta
$$
where $\beta>0$.
Determine the smallest value of $\beta$ for which $y_m \geq 4$. | diff |
|||
$ | page61-10 | 102,965.21 | 102,965.21 | A home buyer can afford to spend no more than $\$ 800 /$ month on mortgage payments. Suppose that the interest rate is $9 \%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.
Determine the total interest paid during the term of the mortgage. | diff |
||
page593-8 | 4 | 4 | Find the fundamental period of the given function:
$$f(x)=\left\{\begin{array}{ll}(-1)^n, & 2 n-1 \leq x<2 n, \\ 1, & 2 n \leq x<2 n+1 ;\end{array} \quad n=0, \pm 1, \pm 2, \ldots\right.$$ | diff |
|||
% | page131-13 | 9.73 | 9.73 | A homebuyer wishes to finance the purchase with a \$95,000 mortgage with a 20-year term. What is the maximum interest rate the buyer can afford if the monthly payment is not to exceed \$900? | diff |
||
$ | page131-10 | 804.62 | 804.62 | A homebuyer wishes to take out a mortgage of $100,000 for a 30-year period. What monthly payment is required if the interest rate is 9%? | diff |
||
${ }^{\circ} \mathrm{C}$ | page619-18 | 35.91 | 35.91 | Let a metallic rod $20 \mathrm{~cm}$ long be heated to a uniform temperature of $100^{\circ} \mathrm{C}$. Suppose that at $t=0$ the ends of the bar are plunged into an ice bath at $0^{\circ} \mathrm{C}$, and thereafter maintained at this temperature, but that no heat is allowed to escape through the lateral surface. Determine the temperature at the center of the bar at time $t=30 \mathrm{~s}$ if the bar is made of silver. | diff |
||
page277-37 | 2 | 2 | Find $\gamma$ so that the solution of the initial value problem $x^2 y^{\prime \prime}-2 y=0, y(1)=1, y^{\prime}(1)=\gamma$ is bounded as $x \rightarrow 0$. | diff |
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Page 60-5 | 0.24995 | 0.24995 | A tank contains 100 gal of water and $50 \mathrm{oz}$ of salt. Water containing a salt concentration of $\frac{1}{4}\left(1+\frac{1}{2} \sin t\right) \mathrm{oz} / \mathrm{gal}$ flows into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, and the mixture in the tank flows out at the same rate.
The long-time behavior of the solution is an oscillation about a certain constant level. What is the amplitude of the oscillation? | diff |
|||
$\mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}$ | page203-17 | 8 | 8 | A mass weighing $8 \mathrm{lb}$ stretches a spring 1.5 in. The mass is also attached to a damper with coefficient $\gamma$. Determine the value of $\gamma$ for which the system is critically damped; be sure to give the units for $\gamma$ | diff |
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hour | Page 18 19 | 7.136 | 7.136 | Your swimming pool containing 60,000 gal of water has been contaminated by $5 \mathrm{~kg}$ of a nontoxic dye that leaves a swimmer's skin an unattractive green. The pool's filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of $200 \mathrm{gal} / \mathrm{min}$. Find the time $T$ at which the concentration of dye first reaches the value $0.02 \mathrm{~g} / \mathrm{gal}$. | diff |
||
1 25(c) | $\frac{2}{49}$ | 0.0408 | For small, slowly falling objects, the assumption made in the text that the drag force is proportional to the velocity is a good one. For larger, more rapidly falling objects, it is more accurate to assume that the drag force is proportional to the square of the velocity. If m = 10 kg, find the drag coefficient so that the limiting velocity is 49 m/s. | diff |
|||
page163-23 | 10.7598 | 10.7598 | Consider the initial value problem
$$
3 u^{\prime \prime}-u^{\prime}+2 u=0, \quad u(0)=2, \quad u^{\prime}(0)=0
$$
For $t>0$ find the first time at which $|u(t)|=10$. | diff |
|||
page172-18 | 1.5 | 1.5 | Consider the initial value problem
$$
9 y^{\prime \prime}+12 y^{\prime}+4 y=0, \quad y(0)=a>0, \quad y^{\prime}(0)=-1
$$
Find the critical value of $a$ that separates solutions that become negative from those that are always positive. | diff |
|||
$\$$ | 2.3.2 | 588313 | 588313 | For instance, suppose that one opens an individual retirement account (IRA) at age 25 and makes annual investments of $\$ 2000$ thereafter in a continuous manner. Assuming a rate of return of $8 \%$, what will be the balance in the IRA at age 65 ? |
We have $S_0=0, r=0.08$, and $k=\$ 2000$, and we wish to determine $S(40)$. From Eq. $$
S(t)=S_0 e^{r t}+(k / r)\left(e^{r t}-1\right)
$$ we have
$$
S(40)=(25,000)\left(e^{3.2}-1\right)=\$ 588,313
$$
| diff |
|
$\mathrm{ft}$ | 3.7.2 | 0.18162 | 0.18162 | Suppose that a mass weighing $10 \mathrm{lb}$ stretches a spring $2 \mathrm{in}$. If the mass is displaced an additional 2 in. and is then set in motion with an initial upward velocity of $1 \mathrm{ft} / \mathrm{s}$, by determining the position of the mass at any later time, calculate the amplitude of the motion. |
The spring constant is $k=10 \mathrm{lb} / 2 \mathrm{in} .=60 \mathrm{lb} / \mathrm{ft}$, and the mass is $m=w / g=10 / 32 \mathrm{lb} \cdot \mathrm{s}^2 / \mathrm{ft}$. Hence the equation of motion reduces to
$$
u^{\prime \prime}+192 u=0
$$
and the general solution is
$$
u=A \cos (8 \sqrt{3} t)+B \sin (8 \sqrt{3} t)
$$
The solution satisfying the initial conditions $u(0)=1 / 6 \mathrm{ft}$ and $u^{\prime}(0)=-1 \mathrm{ft} / \mathrm{s}$ is
$$
u=\frac{1}{6} \cos (8 \sqrt{3} t)-\frac{1}{8 \sqrt{3}} \sin (8 \sqrt{3} t)
$$
The natural frequency is $\omega_0=\sqrt{192} \cong 13.856 \mathrm{rad} / \mathrm{s}$, so the period is $T=2 \pi / \omega_0 \cong 0.45345 \mathrm{~s}$. The amplitude $R$ and phase $\delta$ are found from Eqs. $$
R=\sqrt{A^2+B^2}, \quad \tan \delta=B / A
$$. We have
$$
R^2=\frac{1}{36}+\frac{1}{192}=\frac{19}{576}, \quad \text { so } \quad R \cong 0.18162 \mathrm{ft}
$$
| diff |
|
$\mathrm{~min}$ | 2.3.1 | $(\ln 50) / 0.03$ | 130.400766848 | At time $t=0$ a tank contains $Q_0 \mathrm{lb}$ of salt dissolved in 100 gal of water. Assume that water containing $\frac{1}{4} \mathrm{lb}$ of salt/gal is entering the tank at a rate of $r \mathrm{gal} / \mathrm{min}$ and that the well-stirred mixture is draining from the tank at the same rate. Set up the initial value problem that describes this flow process. By finding the amount of salt $Q(t)$ in the tank at any time, and the limiting amount $Q_L$ that is present after a very long time, if $r=3$ and $Q_0=2 Q_L$, find the time $T$ after which the salt level is within $2 \%$ of $Q_L$. |
We assume that salt is neither created nor destroyed in the tank. Therefore variations in the amount of salt are due solely to the flows in and out of the tank. More precisely, the rate of change of salt in the tank, $d Q / d t$, is equal to the rate at which salt is flowing in minus the rate at which it is flowing out. In symbols,
$$
\frac{d Q}{d t}=\text { rate in }- \text { rate out }
$$
The rate at which salt enters the tank is the concentration $\frac{1}{4} \mathrm{lb} / \mathrm{gal}$ times the flow rate $r \mathrm{gal} / \mathrm{min}$, or $(r / 4) \mathrm{lb} / \mathrm{min}$. To find the rate at which salt leaves the tankl we need to multiply the concentration of salt in the tank by the rate of outflow, $r \mathrm{gal} / \mathrm{min}$. Since the rates of flow in and out are equal, the volume of water in the tank remains constant at $100 \mathrm{gal}$, and since the mixture is "well-stirred," the concentration throughout the tank is the same, namely, $[Q(t) / 100] \mathrm{lb} / \mathrm{gal}$. Therefore the rate at which salt leaves the tank is $[r Q(t) / 100] \mathrm{lb} / \mathrm{min}$. Thus the differential equation governing this process is
$$
\frac{d Q}{d t}=\frac{r}{4}-\frac{r Q}{100}
$$
The initial condition is
$$
Q(0)=Q_0
$$
Upon thinking about the problem physically, we might anticipate that eventually the mixture originally in the tank will be essentially replaced by the mixture flowing in, whose concentration is $\frac{1}{4} \mathrm{lb} / \mathrm{gal}$. Consequently, we might expect that ultimately the amount of salt in the tank would be very close to $25 \mathrm{lb}$. We can also find the limiting amount $Q_L=25$ by setting $d Q / d t$ equal to zero in the equation and solving the resulting algebraic equation for $Q$. Rewriting the above equation in the standard form for a linear equation, we have
$$
\frac{d Q}{d t}+\frac{r Q}{100}=\frac{r}{4}
$$
Thus the integrating factor is $e^{r t / 100}$ and the general solution is
$$
Q(t)=25+c e^{-r t / 100}
$$
where $c$ is an arbitrary constant. To satisfy the initial condition, we must choose $c=Q_0-25$. Therefore the solution of the initial value problem is
$$
Q(t)=25+\left(Q_0-25\right) e^{-r t / 100}
$$
or
$$
Q(t)=25\left(1-e^{-r t / 100}\right)+Q_0 e^{-r t / 100}
$$
From Eq., you can see that $Q(t) \rightarrow 25$ (lb) as $t \rightarrow \infty$, so the limiting value $Q_L$ is 25 , confirming our physical intuition. Further, $Q(t)$ approaches the limit more rapidly as $r$ increases. In interpreting the solution, note that the second term on the right side is the portion of the original salt that remains at time $t$, while the first term gives the amount of salt in the tank due to the action of the flow processes. Now suppose that $r=3$ and $Q_0=2 Q_L=50$; then Eq. becomes
$$
Q(t)=25+25 e^{-0.03 t}
$$
Since $2 \%$ of 25 is 0.5 , we wish to find the time $T$ at which $Q(t)$ has the value 25.5. Substituting $t=T$ and $Q=25.5$ in Eq. (8) and solving for $T$, we obtain
$$
T=(\ln 50) / 0.03 \cong 130.400766848(\mathrm{~min}) .
$$
| diff |
|
$\mathrm{rad}$ | 3.7.2 | $-\arctan (\sqrt{3} / 4)$ | -0.40864 | Suppose that a mass weighing $10 \mathrm{lb}$ stretches a spring $2 \mathrm{in}$. If the mass is displaced an additional 2 in. and is then set in motion with an initial upward velocity of $1 \mathrm{ft} / \mathrm{s}$, by determining the position of the mass at any later time, calculate the phase of the motion. |
The spring constant is $k=10 \mathrm{lb} / 2 \mathrm{in} .=60 \mathrm{lb} / \mathrm{ft}$, and the mass is $m=w / g=10 / 32 \mathrm{lb} \cdot \mathrm{s}^2 / \mathrm{ft}$. Hence the equation of motion reduces to
$$
u^{\prime \prime}+192 u=0
$$
and the general solution is
$$
u=A \cos (8 \sqrt{3} t)+B \sin (8 \sqrt{3} t)
$$
The solution satisfying the initial conditions $u(0)=1 / 6 \mathrm{ft}$ and $u^{\prime}(0)=-1 \mathrm{ft} / \mathrm{s}$ is
$$
u=\frac{1}{6} \cos (8 \sqrt{3} t)-\frac{1}{8 \sqrt{3}} \sin (8 \sqrt{3} t)
$$
The natural frequency is $\omega_0=\sqrt{192} \cong 13.856 \mathrm{rad} / \mathrm{s}$, so the period is $T=2 \pi / \omega_0 \cong 0.45345 \mathrm{~s}$. The amplitude $R$ and phase $\delta$ are found from Eqs. (17) $$R=\sqrt{A^2+B^2}, \quad \tan \delta=B / A$$. We have
$$
R^2=\frac{1}{36}+\frac{1}{192}=\frac{19}{576}, \quad \text { so } \quad R \cong 0.18162 \mathrm{ft}
$$
The second of Eqs. (17) yields $\tan \delta=-\sqrt{3} / 4$. There are two solutions of this equation, one in the second quadrant and one in the fourth. In the present problem $\cos \delta>0$ and $\sin \delta<0$, so $\delta$ is in the fourth quadrant, namely,
$$
\delta=-\arctan (\sqrt{3} / 4) \cong-0.40864 \mathrm{rad}
$$ | diff |
|
$10^6 \mathrm{~kg}$ | 2.5.1 | 46.7 | 46.7 | The logistic model has been applied to the natural growth of the halibut population in certain areas of the Pacific Ocean. ${ }^{12}$ Let $y$, measured in kilograms, be the total mass, or biomass, of the halibut population at time $t$. The parameters in the logistic equation are estimated to have the values $r=0.71 /$ year and $K=80.5 \times 10^6 \mathrm{~kg}$. If the initial biomass is $y_0=0.25 K$, find the biomass 2 years later. |
It is convenient to scale the solution (11) $$y=\frac{y_0 K}{y_0+\left(K-y_0\right) e^{-r t}}
$$ to the carrying capacity $K$; thus we write Eq. (11) in the form
$$
\frac{y}{K}=\frac{y_0 / K}{\left(y_0 / K\right)+\left[1-\left(y_0 / K\right)\right] e^{-r t}}
$$
Using the data given in the problem, we find that
$$
\frac{y(2)}{K}=\frac{0.25}{0.25+0.75 e^{-1.42}} \cong 0.5797 .
$$
Consequently, $y(2) \cong 46.7 \times 10^6 \mathrm{~kg}$.
| diff |
|
m | 3.01 | 4.8 | 4.8 | In an orienteering class, you have the goal of moving as far (straight-line distance) from base camp as possible by making three straight-line moves. You may use the following displacements in any order: (a) $\vec{a}, 2.0 \mathrm{~km}$ due east (directly toward the east); (b) $\vec{b}, 2.0 \mathrm{~km} 30^{\circ}$ north of east (at an angle of $30^{\circ}$ toward the north from due east); (c) $\vec{c}, 1.0 \mathrm{~km}$ due west. Alternatively, you may substitute either $-\vec{b}$ for $\vec{b}$ or $-\vec{c}$ for $\vec{c}$. What is the greatest distance you can be from base camp at the end of the third displacement? (We are not concerned about the direction.) | fund |
||
$\mathrm{m} / \mathrm{s}^2$ | 4.06 | 83.81 | 83.81 | "Top gun" pilots have long worried about taking a turn too tightly. As a pilot's body undergoes centripetal acceleration, with the head toward the center of curvature, the blood pressure in the brain decreases, leading to loss of brain function.
There are several warning signs. When the centripetal acceleration is $2 g$ or $3 g$, the pilot feels heavy. At about $4 g$, the pilot's vision switches to black and white and narrows to "tunnel vision." If that acceleration is sustained or increased, vision ceases and, soon after, the pilot is unconscious - a condition known as $g$-LOC for " $g$-induced loss of consciousness."
What is the magnitude of the acceleration, in $g$ units, of a pilot whose aircraft enters a horizontal circular turn with a velocity of $\vec{v}_i=(400 \hat{\mathrm{i}}+500 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ and $24.0 \mathrm{~s}$ later leaves the turn with a velocity of $\vec{v}_f=(-400 \hat{\mathrm{i}}-500 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ ? | fund |
||
$10^6$ m | 1.01 | 2 | 2 | The world’s largest ball of string is about 2 m in radius. To the nearest order of magnitude, what is the total length L of the string in the ball? | fund |
||
km | 2.01 | 10.4 | 10.4 | You drive a beat-up pickup truck along a straight road for $8.4 \mathrm{~km}$ at $70 \mathrm{~km} / \mathrm{h}$, at which point the truck runs out of gasoline and stops. Over the next $30 \mathrm{~min}$, you walk another $2.0 \mathrm{~km}$ farther along the road to a gasoline station.
What is your overall displacement from the beginning of your drive to your arrival at the station? | fund |
||
$10^3 \mathrm{~kg} / \mathrm{m}^3$ | 1.02 | 1.4 | 1.4 | A heavy object can sink into the ground during an earthquake if the shaking causes the ground to undergo liquefaction, in which the soil grains experience little friction as they slide over one another. The ground is then effectively quicksand. The possibility of liquefaction in sandy ground can be predicted in terms of the void ratio $e$ for a sample of the ground:
$$
e=\frac{V_{\text {voids }}}{V_{\text {grains }}} .
$$
Here, $V_{\text {grains }}$ is the total volume of the sand grains in the sample and $V_{\text {voids }}$ is the total volume between the grains (in the voids). If $e$ exceeds a critical value of 0.80 , liquefaction can occur during an earthquake. What is the corresponding sand density $\rho_{\text {sand }}$ ? Solid silicon dioxide (the primary component of sand) has a density of $\rho_{\mathrm{SiO}_2}=2.600 \times 10^3 \mathrm{~kg} / \mathrm{m}^3$. | fund |
||
$^{\circ}$ | 3.05 | 109 | 109 | What is the angle $\phi$ between $\vec{a}=3.0 \hat{\mathrm{i}}-4.0 \hat{\mathrm{j}}$ and $\vec{b}=$ $-2.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{k}}$ ? | fund |
||
J | 7.03 | 4.0 | 4.0 | During a storm, a crate of crepe is sliding across a slick, oily parking lot through a displacement $\vec{d}=(-3.0 \mathrm{~m}) \hat{\mathrm{i}}$ while a steady wind pushes against the crate with a force $\vec{F}=(2.0 \mathrm{~N}) \hat{\mathrm{i}}+(-6.0 \mathrm{~N}) \hat{\mathrm{j}}$. If the crate has a kinetic energy of $10 \mathrm{~J}$ at the beginning of displacement $\vec{d}$, what is its kinetic energy at the end of $\vec{d}$ ? | fund |
||
J | 7.08 | 7.0 | 7.0 | When the force on an object depends on the position of the object, we cannot find the work done by it on the object by simply multiplying the force by the displacement. The reason is that there is no one value for the force-it changes. So, we must find the work in tiny little displacements and then add up all the work results. We effectively say, "Yes, the force varies over any given tiny little displacement, but the variation is so small we can approximate the force as being constant during the displacement." Sure, it is not precise, but if we make the displacements infinitesimal, then our error becomes infinitesimal and the result becomes precise. But, to add an infinite number of work contributions by hand would take us forever, longer than a semester. So, we add them up via an integration, which allows us to do all this in minutes (much less than a semester).
Force $\vec{F}=\left(3 x^2 \mathrm{~N}\right) \hat{\mathrm{i}}+(4 \mathrm{~N}) \hat{\mathrm{j}}$, with $x$ in meters, acts on a particle, changing only the kinetic energy of the particle. How much work is done on the particle as it moves from coordinates $(2 \mathrm{~m}, 3 \mathrm{~m})$ to $(3 \mathrm{~m}, 0 \mathrm{~m})$ ? | fund |
||
$10^{42}$ | Question 21.75 | $4.16$ | 4.16 | The charges of an electron and a positron are $-e$ and $+e$. The mass of each is $9.11 \times 10^{-31} \mathrm{~kg}$. What is the ratio of the electrical force to the gravitational force between an electron and a positron?
| fund |
||
$\mathrm{~cm}$ | Question 21.19 | $3.00$ | 3.00 | Particle 1 of charge $+q$ and particle 2 of charge $+4.00 q$ are held at separation $L=9.00 \mathrm{~cm}$ on an $x$ axis. If particle 3 of charge $q_3$ is to be located such that the three particles remain in place when released, what must be the $x$ coordinate of particle 3? | fund |
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$\mathrm{~cm}$ | Question 22.11 | $-30$ | -30 | Two charged particles are fixed to an $x$ axis: Particle 1 of charge $q_1=2.1 \times 10^{-8} \mathrm{C}$ is at position $x=20 \mathrm{~cm}$ and particle 2 of charge $q_2=-4.00 q_1$ is at position $x=70 \mathrm{~cm}$. At what coordinate on the axis (other than at infinity) is the net electric field produced by the two particles equal to zero?
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$\mathrm{fC} $ | Question 23.53 | $7.78$ | 7.78 | The volume charge density of a solid nonconducting sphere of radius $R=5.60 \mathrm{~cm}$ varies with radial distance $r$ as given by $\rho=$ $\left(14.1 \mathrm{pC} / \mathrm{m}^3\right) r / R$. What is the sphere's total charge? | fund |
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$10^4 \mathrm{~N} / \mathrm{C}$ | Question 23.45 | $2.50$ | 2.50 | Two charged concentric spherical shells have radii $10.0 \mathrm{~cm}$ and $15.0 \mathrm{~cm}$. The charge on the inner shell is $4.00 \times 10^{-8} \mathrm{C}$, and that on the outer shell is $2.00 \times 10^{-8} \mathrm{C}$. Find the electric field at $r=12.0 \mathrm{~cm}$. | fund |
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$10^{-10} \mathrm{~N}$ | Question 22.51 | $2.6$ | 2.6 | Assume that a honeybee is a sphere of diameter 1.000 $\mathrm{cm}$ with a charge of $+45.0 \mathrm{pC}$ uniformly spread over its surface. Assume also that a spherical pollen grain of diameter $40.0 \mu \mathrm{m}$ is electrically held on the surface of the bee because the bee's charge induces a charge of $-1.00 \mathrm{pC}$ on the near side of the grain and a charge of $+1.00 \mathrm{pC}$ on the far side. What is the magnitude of the net electrostatic force on the grain due to the bee? | fund |
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$10^2 \mathrm{~N}$ | Question 21.69 | $5.1$ | 5.1 | In the radioactive decay of Eq. 21-13, $\mathrm{a}^{238} \mathrm{U}$ nucleus transforms to ${ }^{234} \mathrm{Th}$ and an ejected ${ }^4 \mathrm{He}$. (These are nuclei, not atoms, and thus electrons are not involved.) When the separation between ${ }^{234} \mathrm{Th}$ and ${ }^4 \mathrm{He}$ is $9.0 \times 10^{-15} \mathrm{~m}$, what are the magnitudes of the electrostatic force between them?
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$\mathrm{~cm}$ | Question 22.73 | $-1.0$ | -1.0 | The electric field in an $x y$ plane produced by a positively charged particle is $7.2(4.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}) \mathrm{N} / \mathrm{C}$ at the point $(3.0,3.0) \mathrm{cm}$ and $100 \hat{\mathrm{i}} \mathrm{N} / \mathrm{C}$ at the point $(2.0,0) \mathrm{cm}$. What is the $x$ coordinate of the particle? | fund |
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$\mathrm{~N} / \mathrm{C}$ | Question 22.69 | $47$ | 47 | Two particles, each with a charge of magnitude $12 \mathrm{nC}$, are at two of the vertices of an equilateral triangle with edge length $2.0 \mathrm{~m}$. What is the magnitude of the electric field at the third vertex if both charges are positive? | fund |
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$10^{-23} \mathrm{~J}$ | Question 22.59 | $1.22$ | 1.22 | How much work is required to turn an electric dipole $180^{\circ}$ in a uniform electric field of magnitude $E=46.0 \mathrm{~N} / \mathrm{C}$ if the dipole moment has a magnitude of $p=3.02 \times$ $10^{-25} \mathrm{C} \cdot \mathrm{m}$ and the initial angle is $64^{\circ} ?$
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$10^8 \mathrm{~N}$ | Question 21.57 | $1.7$ | 1.7 | We know that the negative charge on the electron and the positive charge on the proton are equal. Suppose, however, that these magnitudes differ from each other by $0.00010 \%$. With what force would two copper coins, placed $1.0 \mathrm{~m}$ apart, repel each other? Assume that each coin contains $3 \times 10^{22}$ copper atoms. (Hint: A neutral copper atom contains 29 protons and 29 electrons.) | fund |
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m | Question 21.3 | 1.39 | 1.39 | What must be the distance between point charge $q_1=$ $26.0 \mu \mathrm{C}$ and point charge $q_2=-47.0 \mu \mathrm{C}$ for the electrostatic force between them to have a magnitude of $5.70 \mathrm{~N}$ ?
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$\mathrm{~N} \hat{\mathrm{i}}$ | Question 21.61 | $(0.829)$ | 0.829 | Three charged particles form a triangle: particle 1 with charge $Q_1=80.0 \mathrm{nC}$ is at $x y$ coordinates $(0,3.00 \mathrm{~mm})$, particle 2 with charge $Q_2$ is at $(0,-3.00 \mathrm{~mm})$, and particle 3 with charge $q=18.0$ $\mathrm{nC}$ is at $(4.00 \mathrm{~mm}, 0)$. In unit-vector notation, what is the electrostatic force on particle 3 due to the other two particles if $Q_2$ is equal to $80.0 \mathrm{nC}$?
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$\mathrm{~mm}$ | Question 22.77 | $6.0$ | 6.0 | A particle of charge $-q_1$ is at the origin of an $x$ axis. At what location on the axis should a particle of charge $-4 q_1$ be placed so that the net electric field is zero at $x=2.0 \mathrm{~mm}$ on the axis? | fund |
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$10^{15} \mathrm{~m} / \mathrm{s}^2$ | Question 22.43 | $3.51$ | 3.51 | An electron is released from rest in a uniform electric field of magnitude $2.00 \times 10^4 \mathrm{~N} / \mathrm{C}$. Calculate the acceleration of the electron. (Ignore gravitation.) | fund |
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$10^{21} \mathrm{~N} / \mathrm{C}$ | Question 22.3 | $3.07$ | 3.07 | The nucleus of a plutonium-239 atom contains 94 protons. Assume that the nucleus is a sphere with radius $6.64 \mathrm{fm}$ and with the charge of the protons uniformly spread through the sphere. At the surface of the nucleus, what are the magnitude of the electric field produced by the protons? | fund |
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$10^{-8} \mathrm{C}$ | Question 21.21 | $3.8$ | 3.8 | A nonconducting spherical shell, with an inner radius of $4.0 \mathrm{~cm}$ and an outer radius of $6.0 \mathrm{~cm}$, has charge spread nonuniformly through its volume between its inner and outer surfaces. The volume charge density $\rho$ is the charge per unit volume, with the unit coulomb per cubic meter. For this shell $\rho=b / r$, where $r$ is the distance in meters from the center of the shell and $b=3.0 \mu \mathrm{C} / \mathrm{m}^2$. What is the net charge in the shell?
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$\mathrm{~mm}$ | Question 23.41 | $0.44$ | 0.44 | An electron is shot directly
Figure 23-50 Problem 40. toward the center of a large metal plate that has surface charge density $-2.0 \times 10^{-6} \mathrm{C} / \mathrm{m}^2$. If the initial kinetic energy of the electron is $1.60 \times 10^{-17} \mathrm{~J}$ and if the electron is to stop (due to electrostatic repulsion from the plate) just as it reaches the plate, how far from the plate must the launch point be? | fund |
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$10^7 \mathrm{~N} / \mathrm{C}$ | Question 23.37 | $5.4$ | 5.4 | A square metal plate of edge length $8.0 \mathrm{~cm}$ and negligible thickness has a total charge of $6.0 \times 10^{-6} \mathrm{C}$. Estimate the magnitude $E$ of the electric field just off the center of the plate (at, say, a distance of $0.50 \mathrm{~mm}$ from the center) by assuming that the charge is spread uniformly over the two faces of the plate. | fund |
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$N$ | Question 21.49 | $3.8$ | 3.8 | A neutron consists of one "up" quark of charge $+2 e / 3$ and two "down" quarks each having charge $-e / 3$. If we assume that the down quarks are $2.6 \times 10^{-15} \mathrm{~m}$ apart inside the neutron, what is the magnitude of the electrostatic force between them?
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$10^6 \mathrm{~m} / \mathrm{s}$ | Question 21.73 | $2.19$ | 2.19 | In an early model of the hydrogen atom (the Bohr model), the electron orbits the proton in uniformly circular motion. The radius of the circle is restricted (quantized) to certain values given by where $a_0=52.92 \mathrm{pm}$. What is the speed of the electron if it orbits in the smallest allowed orbit? | fund |
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$\mathrm{~m}$ | Question 22.35 | $0.346$ | 0.346 | At what distance along the central perpendicular axis of a uniformly charged plastic disk of radius $0.600 \mathrm{~m}$ is the magnitude of the electric field equal to one-half the magnitude of the field at the center of the surface of the disk? | fund |
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Question 21.55 | $0.5$ | 0.5 | Of the charge $Q$ on a tiny sphere, a fraction $\alpha$ is to be transferred to a second, nearby sphere. The spheres can be treated as particles. What value of $\alpha$ maximizes the magnitude $F$ of the electrostatic force between the two spheres? | fund |
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Question 21.71 | $0$ | 0 | In a spherical metal shell of radius $R$, an electron is shot from the center directly toward a tiny hole in the shell, through which it escapes. The shell is negatively charged with a surface charge density (charge per unit area) of $6.90 \times 10^{-13} \mathrm{C} / \mathrm{m}^2$. What is the magnitude of the electron's acceleration when it reaches radial distances $r=0.500 R$? | fund |
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N | Question 21.5 | 2.81 | 2.81 | A particle of charge $+3.00 \times 10^{-6} \mathrm{C}$ is $12.0 \mathrm{~cm}$ distant from a second particle of charge $-1.50 \times 10^{-6} \mathrm{C}$. Calculate the magnitude of the electrostatic force between the particles. | fund |
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$\mathrm{pC}$ | Question 22.5 | $56$ | 56 | A charged particle produces an electric field with a magnitude of $2.0 \mathrm{~N} / \mathrm{C}$ at a point that is $50 \mathrm{~cm}$ away from the particle. What is the magnitude of the particle's charge?
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$e$ | Question 22.39 | $-5$ | -5 | In Millikan's experiment, an oil drop of radius $1.64 \mu \mathrm{m}$ and density $0.851 \mathrm{~g} / \mathrm{cm}^3$ is suspended in chamber C when a downward electric field of $1.92 \times 10^5 \mathrm{~N} / \mathrm{C}$ is applied. Find the charge on the drop, in terms of $e$. | fund |
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$\mathrm{~N}$ | Question 21.15 | $35$ | 35 | The charges and coordinates of two charged particles held fixed in an $x y$ plane are $q_1=+3.0 \mu \mathrm{C}, x_1=3.5 \mathrm{~cm}, y_1=0.50 \mathrm{~cm}$, and $q_2=-4.0 \mu \mathrm{C}, x_2=-2.0 \mathrm{~cm}, y_2=1.5 \mathrm{~cm}$. Find the magnitude of the electrostatic force on particle 2 due to particle 1. | fund |
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$10^{-15} \mathrm{~N}$ | Question 22.45 | $6.6$ | 6.6 | An electron on the axis of an electric dipole is $25 \mathrm{~nm}$ from the center of the dipole. What is the magnitude of the electrostatic force on the electron if the dipole moment is $3.6 \times 10^{-29} \mathrm{C} \cdot \mathrm{m}$ ? Assume that $25 \mathrm{~nm}$ is much larger than the separation of the charged particles that form the dipole. | fund |
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$\mathrm{~mA}$ | Question 21.31 | $122$ | 122 | Earth's atmosphere is constantly bombarded by cosmic ray protons that originate somewhere in space. If the protons all passed through the atmosphere, each square meter of Earth's surface would intercept protons at the average rate of 1500 protons per second. What would be the electric current intercepted by the total surface area of the planet? | fund |
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$C$ | Question 22.81 | $-0.029$ | -0.029 | An electric field $\vec{E}$ with an average magnitude of about $150 \mathrm{~N} / \mathrm{C}$ points downward in the atmosphere near Earth's surface. We wish to "float" a sulfur sphere weighing $4.4 \mathrm{~N}$ in this field by charging the sphere. What charge (both sign and magnitude) must be used? | fund |
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$ \mu \mathrm{C}$ | Question 21.9 | $-1.00 \mu \mathrm{C}$ | -1.00 | Two identical conducting spheres, fixed in place, attract each other with an electrostatic force of $0.108 \mathrm{~N}$ when their center-to-center separation is $50.0 \mathrm{~cm}$. The spheres are then connected by a thin conducting wire. When the wire is removed, the spheres repel each other with an electrostatic force of $0.0360 \mathrm{~N}$. Of the initial charges on the spheres, with a positive net charge, what was the negative charge on one of them? | fund |
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$10^6$ | Question 22.55 | $2.7$ | 2.7 | A uniform electric field exists in a region between two oppositely charged plates. An electron is released from rest at the surface of the negatively charged plate and strikes the surface of the opposite plate, $2.0 \mathrm{~cm}$ away, in a time $1.5 \times 10^{-8} \mathrm{~s}$. What is the speed of the electron as it strikes the second plate? | fund |
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$10^{-6} \mathrm{~kg}$ | Question 21.63 | $2.2$ | 2.2 | Two point charges of $30 \mathrm{nC}$ and $-40 \mathrm{nC}$ are held fixed on an $x$ axis, at the origin and at $x=72 \mathrm{~cm}$, respectively. A particle with a charge of $42 \mu \mathrm{C}$ is released from rest at $x=28 \mathrm{~cm}$. If the initial acceleration of the particle has a magnitude of $100 \mathrm{~km} / \mathrm{s}^2$, what is the particle's mass? | fund |
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$L$ | Question 21.67 | $2.72$ | 2.72 | In Fig. 21-26, particle 1 of charge $-5.00 q$ and particle 2 of charge $+2.00 q$ are held at separation $L$ on an $x$ axis. If particle 3 of unknown charge $q_3$ is to be located such that the net electrostatic force on it from particles 1 and 2 is zero, what must be the $x$ coordinate of particle 3? | fund |
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$10^{-6} \mathrm{C} $ | Question 23.21 | $-3.0$ | -3.0 | An isolated conductor has net charge $+10 \times 10^{-6} \mathrm{C}$ and a cavity with a particle of charge $q=+3.0 \times 10^{-6} \mathrm{C}$. What is the charge on the cavity wall? | fund |
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$\mu \mathrm{C}$ | Question 21.47 | $-45$ | -45 | Point charges of $+6.0 \mu \mathrm{C}$ and $-4.0 \mu \mathrm{C}$ are placed on an $x$ axis, at $x=8.0 \mathrm{~m}$ and $x=16 \mathrm{~m}$, respectively. What charge must be placed at $x=24 \mathrm{~m}$ so that any charge placed at the origin would experience no electrostatic force?
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$\mu \mathrm{C}$ | Question 23.13 | $3.54$ | 3.54 | The electric field in a certain region of Earth's atmosphere is directed vertically down. At an altitude of $300 \mathrm{~m}$ the field has magnitude $60.0 \mathrm{~N} / \mathrm{C}$; at an altitude of $200 \mathrm{~m}$, the magnitude is $100 \mathrm{~N} / \mathrm{C}$. Find the net amount of charge contained in a cube $100 \mathrm{~m}$ on edge, with horizontal faces at altitudes of 200 and $300 \mathrm{~m}$. | fund |
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$10^9 \mathrm{~N}$ | Question 21.53 | $8.99$ | 8.99 | What would be the magnitude of the electrostatic force between two 1.00 C point charges separated by a distance of $1.00 \mathrm{~m}$ if such point charges existed (they do not) and this configuration could be set up? | fund |
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$10^{-15} \mathrm{C} \cdot \mathrm{m}$ | Question 22.57 | $9.30$ | 9.30 | An electric dipole consisting of charges of magnitude $1.50 \mathrm{nC}$ separated by $6.20 \mu \mathrm{m}$ is in an electric field of strength 1100 $\mathrm{N} / \mathrm{C}$. What is the magnitude of the electric dipole moment? | fund |
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$10^{13} \mathrm{C}$ | Question 21.41 | $5.7$ | 5.7 | What equal positive charges would have to be placed on Earth and on the Moon to neutralize their gravitational attraction? | fund |
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$10^{-19} \mathrm{~N}$ | Question 21.65 | $4.68$ | 4.68 | The initial charges on the three identical metal spheres in Fig. 21-24 are the following: sphere $A, Q$; sphere $B,-Q / 4$; and sphere $C, Q / 2$, where $Q=2.00 \times 10^{-14}$ C. Spheres $A$ and $B$ are fixed in place, with a center-to-center separation of $d=1.20 \mathrm{~m}$, which is much larger than the spheres. Sphere $C$ is touched first to sphere $A$ and then to sphere $B$ and is then removed. What then is the magnitude of the electrostatic force between spheres $A$ and $B$ ?
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$\mathrm{~N}$ | Question 22.49 | $0.245$ | 0.245 | A $10.0 \mathrm{~g}$ block with a charge of $+8.00 \times 10^{-5} \mathrm{C}$ is placed in an electric field $\vec{E}=(3000 \hat{\mathrm{i}}-600 \hat{\mathrm{j}}) \mathrm{N} / \mathrm{C}$. What is the magnitude of the electrostatic force on the block? | fund |
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$10^6 \mathrm{~N} / \mathrm{C} $ | Question 23.31 | $2.3$ | 2.3 | Two long, charged, thin-walled, concentric cylindrical shells have radii of 3.0 and $6.0 \mathrm{~cm}$. The charge per unit length is $5.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ on the inner shell and $-7.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ on the outer shell. What is the magnitude $E$ of the electric field at radial distance $r=4.0 \mathrm{~cm}$? | fund |
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$10^5 \mathrm{~N} \cdot \mathrm{m}^2 / \mathrm{C}$ | Question 23.7 | $2.0$ | 2.0 | A particle of charge $1.8 \mu \mathrm{C}$ is at the center of a Gaussian cube $55 \mathrm{~cm}$ on edge. What is the net electric flux through the surface? | fund |
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$\mu C$ | Question 23.23 | $0.32$ | 0.32 | The drum of a photocopying machine has a length of $42 \mathrm{~cm}$ and a diameter of $12 \mathrm{~cm}$. The electric field just above the drum's surface is $2.3 \times 10^5 \mathrm{~N} / \mathrm{C}$. What is the total charge on the drum? | fund |
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$10^{-15} \mathrm{~N} $ | Question 22.63 | $8.87$ | 8.87 | A spherical water drop $1.20 \mu \mathrm{m}$ in diameter is suspended in calm air due to a downward-directed atmospheric electric field of magnitude $E=462 \mathrm{~N} / \mathrm{C}$. What is the magnitude of the gravitational force on the drop? | fund |
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$10^{11}$ | Question 21.25 | $6.3$ | 6.3 | How many electrons would have to be removed from a coin to leave it with a charge of $+1.0 \times 10^{-7} \mathrm{C}$ ? | fund |
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$\mathrm{nC}$ | Question 23.47 | $-7.5$ | -7.5 | An unknown charge sits on a conducting solid sphere of radius $10 \mathrm{~cm}$. If the electric field $15 \mathrm{~cm}$ from the center of the sphere has the magnitude $3.0 \times 10^3 \mathrm{~N} / \mathrm{C}$ and is directed radially inward, what is the net charge on the sphere?
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$10^{-7} \mathrm{C} / \mathrm{m}^2 $ | Question 23.19 | $4.5$ | 4.5 | Space vehicles traveling through Earth's radiation belts can intercept a significant number of electrons. The resulting charge buildup can damage electronic components and disrupt operations. Suppose a spherical metal satellite $1.3 \mathrm{~m}$ in diameter accumulates $2.4 \mu \mathrm{C}$ of charge in one orbital revolution. Find the resulting surface charge density. | fund |
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$\mathrm{~N} / \mathrm{C}$ | Question 22.71 | $38$ | 38 | A charge of $20 \mathrm{nC}$ is uniformly distributed along a straight rod of length $4.0 \mathrm{~m}$ that is bent into a circular arc with a radius of $2.0 \mathrm{~m}$. What is the magnitude of the electric field at the center of curvature of the arc? | fund |
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$10^7 \mathrm{C}$ | Question 21.33 | $1.3$ | 1.3 | Calculate the number of coulombs of positive charge in 250 $\mathrm{cm}^3$ of (neutral) water. (Hint: A hydrogen atom contains one proton; an oxygen atom contains eight protons.) | fund |
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$10^3 \mathrm{~N} / \mathrm{C}$ | Question 22.41 | $1.5$ | 1.5 | A charged cloud system produces an electric field in the air near Earth's surface. A particle of charge $-2.0 \times 10^{-9} \mathrm{C}$ is acted on by a downward electrostatic force of $3.0 \times 10^{-6} \mathrm{~N}$ when placed in this field. What is the magnitude of the electric field? | fund |
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$10^{-26} \mathrm{~J} $ | Question 22.83 | $-1.49$ | -1.49 | An electric dipole with dipole moment
$$
\vec{p}=(3.00 \hat{\mathrm{i}}+4.00 \hat{\mathrm{j}})\left(1.24 \times 10^{-30} \mathrm{C} \cdot \mathrm{m}\right)
$$
is in an electric field $\vec{E}=(4000 \mathrm{~N} / \mathrm{C}) \hat{\mathrm{i}}$. What is the potential energy of the electric dipole? | fund |