Chapter 26

Find the center of mass (in cm) of the particles with the given masses located at the given points on the \(x\) -axis. $$\text { 5.0 } \mathrm{g} \text { at }(1.0,0), 8.5 \mathrm{g} \text { at }(4.2,0), 3.6 \mathrm{g} \text { at }(7.3,0)$$

Find the volume generated by revolving the region bounded by \(y=4-2 x, x=0,\) and \(y=0\) about the indicated axis, using the indicated element of volume. \(x\) -axis (disks)

The spring of a spring balance is 8.0 in. long when there is no weight on the balance, and it is 9.5 in. long with 6.0 lb hung from the balance. How much work is done in stretching it from 8.0 in. to a length of 10.0 in?

Find the moment of inertia (in \(\mathrm{g} \cdot \mathrm{cm}^{2}\) ) and the radius of gyration (in \(\mathrm{cm}\) ) with respect to the origin of each of the given arrays of masses located at the given points on the \(x\) -axis. $$4.2 \mathrm{g} \text { at }(1.7,0), 3.2 \mathrm{g} \text { at }(3.5,0)$$

Find the areas bounded by the indicated curves. $$y=4 x, y=0, x=1$$

What is the velocity (in \(\mathrm{ft} / \mathrm{s}\) ) of a sandbag \(1.5 \mathrm{s}\) after it is released from a hot-air balloon that is rising at \(12 \mathrm{ft} / \mathrm{s} ?\) (Hint: The acceleration of gravity is \(-32 \mathrm{ft} / \mathrm{s}^{2}\).)

Find the center of mass (in cm) of the particles with the given masses located at the given points on the \(x\) -axis. $$2.3 \mathrm{g} \text { at }(1.3,0), 6.5 \mathrm{g} \text { at }(5.8,0), 1.2 \mathrm{g} \text { at }(9.5,0)$$

Find the volume generated by revolving the region bounded by \(y=4-2 x, x=0,\) and \(y=0\) about the indicated axis, using the indicated element of volume. \(y\) -axis (disks)

Find the moment of inertia (in \(\mathrm{g} \cdot \mathrm{cm}^{2}\) ) and the radius of gyration (in \(\mathrm{cm}\) ) with respect to the origin of each of the given arrays of masses located at the given points on the \(x\) -axis. $$3.4 \mathrm{g} \text { at }(-1.5,0), 6.0 \mathrm{g} \text { at }(2.1,0), 2.6 \mathrm{g} \text { at }(3.8,0)$$

Find the areas bounded by the indicated curves. $$y=3 x+3, y=0, x=2$$

A beach ball is rolled up a shallow slope with an initial velocity of \(18 \mathrm{ft} / \mathrm{s}\). If the acceleration of the ball is \(3.0 \mathrm{ft} / \mathrm{s}^{2}\) down the slope, find the velocity of the ball after \(8.0 \mathrm{s}\)

Find the center of mass (in cm) of the particles with the given masses located at the given points on the \(x\) -axis. $$\text { 31 } \mathrm{g} \text { at }(-3.5,0), 24 \mathrm{g} \text { at }(0,0), 15 \mathrm{g} \text { at }(2.6,0), 84 \mathrm{g} \text { at }(3.7,0)$$

Find the volume generated by revolving the region bounded by \(y=4-2 x, x=0,\) and \(y=0\) about the indicated axis, using the indicated element of volume. \(y\) -axis (shells)

A 160 -lb person compresses a bathroom scale 0.080 in. If the scale obeys Hooke's law, how much work is done compressing the scale if a 180 -lb person stands on it?

Find the moment of inertia (in \(\mathrm{g} \cdot \mathrm{cm}^{2}\) ) and the radius of gyration (in \(\mathrm{cm}\) ) with respect to the origin of each of the given arrays of masses located at the given points on the \(x\) -axis. $$82.0 \mathrm{g} \text { at }(-3.80,0), 90.0 \mathrm{g} \text { at }(0.00,0), 62.0 \mathrm{g} \text { at }(5.50,0)$$

Find the areas bounded by the indicated curves. $$y=8-2 x^{2}, y=0$$

A conveyor belt \(8.00 \mathrm{m}\) long moves at \(0.25 \mathrm{m} / \mathrm{s}\). If a package is placed at one end, find its displacement from the other end as a function of time.

Find the center of mass (in cm) of the particles with the given masses located at the given points on the \(x\) -axis. $$\begin{aligned} 550 \mathrm{g} \text { at }(-42,0), 230 \mathrm{g} \text { at }(-27,0), 470 \mathrm{g} \text { at }(16,0), 120 \mathrm{g} \text { at } (22,0) \end{aligned}$$

Find the volume generated by revolving the region bounded by \(y=4-2 x, x=0,\) and \(y=0\) about the indicated axis, using the indicated element of volume. \(x\) -axis (shells)

Find the moment of inertia (in \(\mathrm{g} \cdot \mathrm{cm}^{2}\) ) and the radius of gyration (in \(\mathrm{cm}\) ) with respect to the origin of each of the given arrays of masses located at the given points on the \(x\) -axis. $$\begin{aligned} &564 \mathrm{g} \text { at }(-45.0,0), 326 \mathrm{g} \text { at }(-22.5,0), 720 \mathrm{g} \text { at }(15.4,0), 205 \mathrm{g}\\\ &\text { at }(64.0,0) \end{aligned}$$

During each cycle, the velocity \(v\) (in \(\mathrm{mm} / \mathrm{s}\) ) of a piston is \(v=6 t-6 t^{2},\) where \(t\) is the time (in s). Find the displacement \(s\) of the piston after 0.75 s if the initial displacement is zero.

Find the volume generated by revolving the regions bounded by the given curves about the \(x\) -axis. Use the indicated method in each case. $$y=2 x, y=0, x=3 \quad \text { (disks) }$$

The velocity (in \(\mathrm{km} / \mathrm{h}\) ) of a plane flying into an increasing headwind is \(v=50(12-t),\) where \(t\) is the time (in h). How far does the plane travel in a 2.0 -h trip?

Find the volume generated by revolving the regions bounded by the given curves about the \(x\) -axis. Use the indicated method in each case. $$y=\sqrt{x}, x=0, y=2 \quad \text { (shells) }$$

Find the indicated moment of inertia or radius of gyration. Find the radius of gyration of a plate covering the region bounded by \(x=2, x=4, y=0,\) and \(y=4,\) with respect to the \(y\) -axis.

Find the areas bounded by the indicated curves. $$y=4 x^{2}-6 x, y=0$$

A cyclist goes downhill for 15 min with a velocity to ling \(v=40+64 t(\text { in } \mathrm{km} / \mathrm{h}),\) and then maintains the speed at the bottom for another 30 min. How far does the cyclist go in the 45 min?

Find the volume generated by revolving the regions bounded by the given curves about the \(x\) -axis. Use the indicated method in each case. $$y=3 \sqrt{x}, y=0, x=4 \quad(\text { disks })$$

Find the indicated moment of inertia or radius of gyration. Find the moment of inertia of a plate covering the first-quadrant region bounded by \(y^{2}=x, x=9,\) and the \(x\) -axis with respect to the \(x\) -axis.

The gravitational force (in \(1 \mathrm{b}\) ) of attraction between two objects is given by \(F=k / x^{2},\) where \(x\) is the distance between the objects. If the objects are \(10 \mathrm{ft}\) apart, find the work required to separate them until they are \(100 \mathrm{ft}\) apart. Express the result in terms of \(k\).

Find the areas bounded by the indicated curves. $$y=3 / x^{2}, y=0, x=2, x=3$$

Find the volume generated by revolving the regions bounded by the given curves about the \(x\) -axis. Use the indicated method in each case. $$y=4 x-x^{2}, y=0 \quad \text { (disks) }$$

Find the indicated moment of inertia or radius of gyration. Find the moment of inertia of a plate covering the region bounded by \(y=2 x, x=1, x=2,\) and the \(x\) -axis with respect to the \(y\) -axis.

Find the work done in winding up (a) \(30 \mathrm{m}\) of a \(40-\mathrm{m}\) rope on which the force of gravity is \(6.0 \mathrm{N} / \mathrm{m},\) and \((\mathrm{b})\) all of the rope.

Find the areas bounded by the indicated curves. $$y=16-x^{2}, y=0, x=-2, x=3$$

In designing a highway, a civil engineer must determine the length of a highway on-ramp for cars going onto the ramp at \(25 \mathrm{km} / \mathrm{h}\) and entering the highway at \(95 \mathrm{km} / \mathrm{h}\) in \(12.0 \mathrm{s}\). What minimum length should the on-ramp be?

Find the volume generated by revolving the regions bounded by the given curves about the \(x\) -axis. Use the indicated method in each case. $$y=x^{3}, y=8, x=0 \quad \text { (shells) }$$

A 1500 -lb elevator is suspended on cables that together weigh 12 Ib/ft. How much work is done in raising the elevator from the basement to the top floor, a distance of \(24 \mathrm{ft} ?\)

Find the areas bounded by the indicated curves. $$y=4 \sqrt{x}, x=0, y=1, y=3$$

While in the barrel of a tennis ball machine, the acceleration \(a\) (in \(\left.\mathrm{ft} / \mathrm{s}^{2}\right)\) of a ball is \(a=90 \sqrt{1-4 t},\) where \(t\) is the time (in s). If \(v=0\) for \(t=0,\) find the velocity of the ball as it leaves the barrel at \(t=0.25 \mathrm{s}\)

A chain is being unwound from a winch. The force of gravity on it is \(12.0 \mathrm{N} / \mathrm{m} .\) When \(20 \mathrm{m}\) have been unwound, how much work is done by gravity in unwinding another \(30 \mathrm{m} ?\)

Find the areas bounded by the indicated curves. $$y=3 \sqrt{x+1}, x=0, y=6$$

A person skis down a slope with an acceleration (in \(\mathrm{m} / \mathrm{s}^{2}\) ) given by \(a=\frac{600 t}{\left(60+0.5 t^{2}\right)^{2}},\) where \(t\) is the time (in s). Find the skier's velocity as a function of time if \(v=0\) when \(t=0\)

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by \(y=4-x\) and the axes

Find the volume generated by revolving the regions bounded by the given curves about the \(x\) -axis. Use the indicated method in each case. $$y=x^{2}+1, x=0, x=3, y=0 \quad \text { (disks) }$$

Find the indicated moment of inertia or radius of gyration. Find the radius of gyration of a plate covering the region bounded by \(y=x^{2}, x=3,\) and the \(x\) -axis with respect to the \(x\) -axis.

At liftoff, a rocket weighs 32.5 tons, including the weight of its fuel. During the first (vertical) stage of ascent, fuel is consumed at the rate of 1.25 tons per \(1000 \mathrm{ft}\) of ascent. How much work is done in lifting the rocket to an altitude of \(12,000 \mathrm{ft} ?\)

Find the areas bounded by the indicated curves. $$y=2 / \sqrt{x}, x=0, y=1, y=4$$

A certain Chevrolet Corvette goes from 0 mi/h to \(60.0 \mathrm{mi} / \mathrm{h}\) \((88.0 \mathrm{ft} / \mathrm{s})\) in \(3.60 \mathrm{s}\). Assuming constant acceleration, how far (in \(\mathrm{ft}\) ) does it travel in this time?

Find the volume generated by revolving the regions bounded by the given curves about the \(x\) -axis. Use the indicated method in each case. $$y=6-x-x^{2}, x=0, y=0 \quad \text { (quadrant I), (disks) }$$