Chapter 14

CP SHM in a Car Engine. The motion of the piston of an automobile engine is approximately simple harmonic. (a) If the stroke of an engine (twice the amplitude) is 0.100 \(\mathrm{m}\) and the engine runs at 4500 \(\mathrm{rev} / \mathrm{min}\) , compute the acceleration of the piston at the endpoint of its stroke. (b) If the piston has mass \(0.450 \mathrm{kg},\) what net force must be exerted on it at this point? (c) What are the speed and kinetic energy of the piston at the mid- point of its stroke? (d) What average power is required to accelerate the piston from rest to the speed found in part (c)? (e) If the engine runs at 7000 rev/min, what are the answers to parts (b), (c), and (d)?

Four passengers with combined mass 250 kg compress the springs of a car with worn-out shock absorbers by 4.00 \(\mathrm{cm}\) when they get in. Model the car and passengers as a single body on a single ideal spring. If the loaded car has a period of vibration of 1.92 s, what is the period of vibration of the empty car?

A glider is oscillating in \(\mathrm{SHM}\) on an air track with an amplitude \(A_{1 .}\) You slow it so that its amplitude is halved. What happens to its (a) period, frequency, and angular frequency; (b) total mechanical energy; (c) maximum speed; (d) speed at \(x=\pm A_{1} / 4 ;\) (e) potential and kinetic energies at \(x=\pm A_{1} / 4 ?\)

CP A child with poor table manners is sliding his \(250-\mathrm{g}\) dinner plate back and forth in \(\mathrm{SHM}\) with an amplitude of 0.100 \(\mathrm{m}\) on a horizontal surface. At a point 0.060 \(\mathrm{m}\) away from equilibrium, the speed of the plate is 0.400 \(\mathrm{m} / \mathrm{s}\) . (a) What is the period? (b) What is the displacement when the speed is 0.160 \(\mathrm{m} / \mathrm{s} ?\) (c) In the center of the dinner plate is a 10.0 -g carrot slice. If the carrot slice is just on the verge of slipping at the endpoint of the path, what is the coefficient of static friction between the carrot slice and the plate?

A 1.50 -kg, horizontal, uniform tray is attached to a vertical ideal spring of force constant 185 \(\mathrm{N} / \mathrm{m}\) and a \(275-\mathrm{g}\) metal ball is in the tray. The spring is below the tray, so it can oscillate up and down. The tray is then pushed down to point \(A\) , which is 15.0 \(\mathrm{cm}\) below the equilibrium point, and released from rest. (a) How high above point \(A\) will the tray be when the metal ball leaves the tray? (Hint: This does not occur when the ball and tray reach their maximum speeds.) (b) How much time elapses between releasing the system at point \(A\) and the ball leaving the tray? (c) How fast is the ball moving just as it leaves the tray?

CP A block with mass \(M\) rests on a frictionless surface and is connected to a horizontal spring of force constant \(k\) . The other end of the spring is attached to a wall (Fig. Pl4.72). A second block with mass \(m\) rests on top of the first block. The coefficient of static friction between the blocks is \(\mu_{\mathrm{s}}\) . Find the maximum amplitude of oscillation such that the top block will not slip on the bot tom block.

CP A 10.0 .0 -kg mass is traveling to the right with a speed of 2.00 \(\mathrm{m} / \mathrm{s}\) on a smooth horizontal surface when it collides with and sticks to a second 10.0 -kg mass that is initially at rest but is attached to a light spring with force constant 110.0 \(\mathrm{N} / \mathrm{m}\) . (a) Find the frequency, amplitude, and period of the subsequent oscillations. (b) How long does it take the system to return the first time to the position it had immediately after the collision?

CP A rocket is accelerating upward at 4.00 \(\mathrm{m} / \mathrm{s}^{2}\) from the launchpad on the earth. Inside a small, 1.50 -kg ball hangs from the ceiling by a light, 1.10-m wire. If the ball is displaced \(8.50^{\circ}\) from the vertical and released, find the amplitude and period of the resulting swings of this pendulum.

An apple weighs 1.00 \(\mathrm{N}\) . When you hang it from the end of a long spring of force constant 1.50 \(\mathrm{N} / \mathrm{m}\) and negligible mass, it bounces up and down in SHM. If you stop the bouncing and let the apple swing from side to side through a small angle, the frequency of this simple pendulum is half the bounce frequency. (Because the angle is small, the back- and-forth swings do not cause any appreciable change in the length of the spring.) What is the unstretched length of the spring (with the apple removed?

CP SHM of a Floating Object. An object with height \(h,\) mass \(M,\) and a uniform cross-sectional area \(A\) floats upright in a liquid with density \(\rho\) (a) Calculate the vertical distance from the surface of the liquid to the bottom of the floating object at equilibrium. (b) A downward force with magnitude \(F\) is applied to the top of the object. At the new equilibrium position, how much farther below the surface of the liquid is the bottom of the object than it was in part (a)? (Assume that some of the object remains above the surface of the liquid.) (c) Your result in part (b) shows that if the force is suddenly removed, the object will oscillate up and down in SHM. Calculate the period of this motion in terms of the density \(\rho\) of the liquid, the mass \(M,\) and the cross-sectional area \(A\) of the object. You can ignore the damping due to fluid friction (see Section 14.7\()\) .

An object with mass 0.200 \(\mathrm{kg}\) is acted on by an elastic restoring force with force constant 10.0 \(\mathrm{N} / \mathrm{m}\) . (a) Graph elastic potential energy \(U\) as a function of displacement \(x\) over a range of \(x\) from \(-0.300 \mathrm{m}\) to \(+0.300 \mathrm{m}\) . On your graph, let \(1 \mathrm{cm}=0.05 \mathrm{J}\) vertically and \(1 \mathrm{cm}=0.05 \mathrm{m}\) horizontally. The object is set into oscillation with an initial potential energy of 0.140 \(\mathrm{J}\) and an initial kinetic energy of 0.060 \(\mathrm{J}\) . Answer the following questions by referring to the graph. (b) What is the amplitude of oscillation? (c) What is the potential energy when the displacement is one half the amplitude? (d) At what displacement are the kinetic and potential energies equal? (e) What is the value of the phase angle \(\phi\) if the initial velocity is positive and the initial displacement is negative?

CALC A 2.00 -kg bucket containing 10.0 \(\mathrm{kg}\) of water is hanging from a vertical ideal spring of force constant 125 \(\mathrm{N} / \mathrm{m}\) and oscillating up and down with an amplitude of 3.00 \(\mathrm{cm} .\) Suddenly the bucket springs a leak in the bottom such that water drops out at a steady rate of 2.00 \(\mathrm{g} / \mathrm{s}\) . When the bucket is half full, find (a) the period of oscillation and (b) the rate at which the period is changing with respect to time. Is the period getting longer or shorter? (c) What is the shortest period this system can have?

A 5.00 -kg partridge is suspended from a pear tree by an ideal spring of negligible mass. When the partridge is pulled down 0.100 m below its equilibrium position and released, it vibrates with a period of 4.20 s. (a) What is its speed as it passes through the equilibrium position? (b) What is its acceleration when it is 0.050 \(\mathrm{m}\) above the equilibrium position? (c) When it is moving upward, how much time is required for it to move from a point 0.050 \(\mathrm{m}\) below its equilibrium position to a point 0.050 \(\mathrm{m}\) above it? (d) The motion of the partridge is stopped, and then it is removed from the spring. How much does the spring shorten?

A 0.0200 -kg bolt moves with SHM that has an amplitude of 0.240 \(\mathrm{m}\) and a period of 1.500 \(\mathrm{s}\) . The displacement of the bolt is \(+0.240 \mathrm{m}\) when \(t=0 .\) Compute (a) the displacement of the bolt when \(t=0.500 \mathrm{s} ;(\mathrm{b})\) the magnitude and direction of the force acting on the bolt when \(t=0.500 \mathrm{s} ;\) (c) the minimum time required for the bolt to move from its initial position to the point where \(x=-0.180 \mathrm{m} ;\) (d) the speed of the bolt when \(x=-0.180 \mathrm{m}\) .

CP SHM of a Butcher's Scale. A spring of negligible mass and force constant \(k=400 \mathrm{N} / \mathrm{m}\) is hung vertically, and a 0.200 -kg pan is suspended from its lower end. A butcher drops a 2.2 -kg steak onto the pan from a height of 0.40 \(\mathrm{m}\) . The steak makes a totally inelastic collision with the pan and sets the system into vertical SHM. What are (a) the speed of the pan and steak immediately after the collision; (b) the amplitude of the subsequent motion; (c) the period of that motion?

A uniform beam is suspended horizontally by two identical vertical springs that are attached between the ceiling and each end of the beam. The beam has mass 225 \(\mathrm{kg}\) , and a 175 -kg sack of gravel sits on the middle of it. The beam is oscillating in \(\mathrm{SHM}\) , with an amplitude of 40.0 \(\mathrm{cm}\) and a frequency of 0.600 cycle/s. (a) The sack of gravel falls off the beam when the beam has its maximum upward displacement. What are the frequency and amplitude of the subsequent SHM of the beam? (b) If the gravel instead falls off when the beam has its maximum speed, what are the frequency and amplitude of the subsequent SHM of the beam?

CP On the planet Newtonia, a simple pendulum having a bob with mass 1.25 \(\mathrm{kg}\) and a length of 185.0 \(\mathrm{cm}\) takes 1.42 \(\mathrm{s}\) . when released from rest, to swing through an angle of \(12.5^{\circ}\) , where it again has zero speed. The circumference of Newtonia is measured to be \(51,400 \mathrm{km}\) . What is the mass of the planet Newtonia?

A 40.0 -N force stretches a vertical spring 0.250 \(\mathrm{m}\) . (a) What mass must be suspended from the spring so that the system will oscillare with a period of 1.00 \(\mathrm{s} ?\) (b) If the amplitude of the motion is 0.050 \(\mathrm{m}\) and the period is that specified in part (a), where is the object and in what direction is it moving 0.35 s after it has passed the equilibrium position, moving downward? (c) What force (magnitude and direction) does the spring exert on the object when it is 0.030 m below the equilibrium position, moving upward?

Don't Miss the Boat. While on a visit to Minnesota ("Land of \(10,000\) Lakes"), you sign up to take an excursion around one of the larger lakes. When you go to the dock where the \(1500-\mathrm{kg}\) boat is tied, you find that the boat is bobbing up and down in the waves, executing simple harmonic motion with amplitude 20 \(\mathrm{cm} .\) The boat takes 3.5 \(\mathrm{s}\) s to make one complete up-and-down cycle. When the boat is at its highest point, its deck is at the same height as the stationary dock. As you watch the boat bob up and down, you (mass 60 \(\mathrm{kg}\) ) begin to feel a bit woozy, due in part to the previous night's dinner of lutefisk. As a result, you refuse to board the boat unless the level of the boat's deck is within 10 \(\mathrm{cm}\) of the dock level. How much time do you have to board the boat comfortably during each cycle of up-and-down motion?

CP A rifle bullet with mass 8.00 \(\mathrm{g}\) and initial horizontal velocity 280 \(\mathrm{m} / \mathrm{s}\) strikes and embeds itself in a block with mass 0.992 \(\mathrm{kg}\) that rests on a frictionless surface and is attached to one end of an ideal spring. The other end of the spring is attached to the wall. The impact compresses the spring a maximum distance of 18.0 \(\mathrm{cm}\) . After the impact, the block moves in SHM. Calculate the period of this motion.

CP CALC For a certain oscillator the net force on the body with mass \(m\) is given by \(F_{x}=-c x^{3}\) (a) What is the potential energy function for this oscillator if we take \(U=0\) at \(x=0\) ? (b) One-quarter of a period is the time for the body to move from \(x=0\) to \(x=A .\) Calculate this time and hence the period. [Hint: Begin with \(\mathrm{Eq} .(14.20),\) modified to include the potential-energy function you found in part (a), and solve for the velocity \(v_{x}\) as a function of \(x\) . Then replace \(v_{x}\) with \(d x / d t\) . Separate the variable by writing all factors containing \(x\) on one side and all factors containing \(t\) on the other side so that each side can be integrated. In the \(x\) -integral make the change of variable \(u=x / A .\) The resulting integral can be evaluated by numerical methods on a computer and has the value \(\int_{0}^{1} d u / \sqrt{1-u^{4}}=1.31 .1(\mathrm{c})\) According to the result you obtained in part (b), does the period depend on the amplitude A of the motion? Are the oscillations simple harmonic?

CP CALC An approximation for the potential energy of a KCl molecule is \(U=A\left[\left(R_{0}^{7} / 8 r^{8}\right)-1 / r\right],\) where \(R_{0}=2.67 \times\) \(10^{-10} \mathrm{m}, A=2.31 \times 10^{-28} \mathrm{J} \cdot \mathrm{m},\) and \(r\) is the distance between the two atoms. Using this approximation: (a) Show that the radial component of the force on each atom is \(F_{r}=A\left[\left(R_{0}^{7} / r^{9}\right)-1 / r^{2}\right].\) (b) Show that \(R_{0}\) is the equilibrium separation. (c) Find the minimum potential energy. (d) Use \(r=R_{0}+x\) and the first two terms of the binomial theorem \(\quad\) (Eq. 14.28\()\) to show that \(F_{r} \approx-\left(7 A / R_{0}^{3}\right) x,\) so that the molecule's force constant is \(k=7 A / R_{0}^{3} .\) (e) With both the \(K\) and \(C 1\) atoms vibrating in opposite directions on opposite sides of the molecule's center of mass, \(m_{1} m_{2} /\left(m_{1}+m_{2}\right)=3.06 \times 10^{-26} \mathrm{kg}\) is the mass to use in cal- culating the frequency. Calculate the frequency of small-amplitude vibrations.

CP Two uniform solid spheres, each with mass \(M=0.800 \mathrm{kg}\) and radius \(R=0.0800 \mathrm{m},\) are connected by a short, light rod that is along a diameter of each sphere and are at rest on a horizontal tabletop. A spring with force constant \(k=160 \mathrm{N} / \mathrm{m}\) has one end attached to the wall and the other end attached to a frictionless ring that passes over the rod at the center of mass of the spheres, which is midway between the centers of the two spheres. The spheres are each pulled the same distance from the wall, stretching the spring, and released. There is sufficient friction between the tabletop and the spheres for the spheres to roll without slipping as they move back and forth on the end of the spring. Show that the motion of the center of mass of the spheres is simple harmonic and calculate the period.

CALC A slender, uni- form, metal rod with mass \(M\) is pivoted without friction about an axis through its midpoint and perpendicular to the rod. A horizontal spring with force constant \(k\) is attached to the lower end of the rod, with the other end of the spring attached to a rigid support. If the rod is displaced by a small angle \(\theta\) from the vertical (Fig. \(P 14.97\) ) and released, show that it moves in angular SHM and calculate the period. (Hint: Assume that the angle \(\theta\) is small enough for the approximations \(\sin \theta \approx \theta\) and \(\cos \theta \approx 1\) to be valid. The motion is simple harmonic if \(d^{2} \theta / d t^{2}=-\omega^{2} \theta,\) and the period is then \(T=2 \pi / \omega .\) )

The Silently Ringing Bell Problem. A large bell is hung from a wooden beam so it can swing back and forth with negligible friction. The center of mass of the bell is 0.60 m below the pivot, the bell has mass 34.0 \(\mathrm{kg}\) , and the moment of inertia of the bell about an axis at the pivot is 18.0 \(\mathrm{kg} \cdot \mathrm{m}^{2} .\) The clapper is a small, \(1.8-\mathrm{kg}\) mass attached to one end of a slender rod that has length \(L\) and negligible mass. The other end of the rod is attached to the inside of the bell so it can swing freely about the same axis as the bell. What should be the length \(L\) of the clapper rod for the bell to ring silently-that is, for the period of oscillation for the bell to equal that for the clapper?

CALC A Spring with Mass. The preceding problems in this chapter have assumed that the springs had negligible mass. But of course no spring is completely massless. To find the effect of the spring's mass, consider a spring with mass \(M\) , equilibrium length \(L_{0}\), and spring constant \(k\). When stretched or compressed to a length \(L,\) the potential energy is \(\frac{1}{2} k x^{2},\) where \(x=L-L_{0-}\) (a) Consider a spring, as described above, that has one end fixed and the other end moving with speed \(v\) . Assume that the speed of points along the length of the spring varies linearly with distance \(I\) from the fixed end. Assume also that the mass \(M\) of the spring is distributed uniformly along the length of the spring. Calculate the kinetic energy of the spring in terms of \(M\) and \(v .\) (Hint: Divide the spring into pieces of length \(d l ;\) find the speed of each piece in terms of \(l, v,\) and \(L ;\) find the mass of each piece in terms of \(d l, M,\) and \(L ;\) and integrate from 0 to \(L\) . The result is not \(\frac{1}{2} M v^{2}\) , since not all of the spring moves with the same speed.) (b) Take the time derivative of the conservation of energy equation, Eq. (14.21), for a mass \(m\) moving on the end of a massless. By comparing your results to Eq. (14.8), which defines \(\omega,\) show that the angular frequency of oscillation is \(\omega=\sqrt{k / m}\) (c) Apply the procedure of part (b) to obtain the angular frequency of oscillation \(\omega\) of the spring considered in part (a). If the effective mass \(M^{\prime}\) of the spring is defined by \(\omega=\sqrt{k / M^{\prime}},\) what is \(M^{\prime}\) in terms of \(M ?\)