Atoms in a solid are not stationary, but vibrate about their equilibrium positions. Typically, the frequency of vibration is about f=2.0×1012 Hz, and the amplitude is about 1.1×10–11 m. For a typical atom, what is its (a) maximum speed and (b) maximum acceleration?

Example 3


In Concept Simulation 10.3 you can explore the concepts that are important in this problem. A block of mass m=0.750 kg is fastened to an unstrained horizontal spring whose spring constant is k=82.0 N/m. The block is given a displacement of +0.120 m, where the + sign indicates that the displacement is along the +x axis, and then released from rest. (a) What is the force (magnitude and direction) that the spring exerts on the block just before the block is released? (b) Find the angular frequency w of the resulting oscillatory motion. (c) What is the maximum speed of the block? (d) Determine the magnitude of the maximum acceleration of the block.


A spiral staircase winds up to the top of a tower in an old castle. To measure the height of the tower, a rope is attached to the top of the tower and hung down the center of the staircase. However, nothing is available with which to measure the length of the rope. Therefore, at the bottom of the rope a small object is attached so as to form a simple pendulum that just clears the floor. The period of the pendulum is measured to be 9.2 s. What is the height of the tower?

Example 10


A person bounces up and down on a trampoline, while always staying in contact with it. The motion is simple harmonic motion, and it takes 1.90 s to complete one cycle. The height of each bounce above the equilibrium position is 45.0 cm. Determine (a) the amplitude and (b) the angular frequency of the motion. (c) What is the maximum speed attained by the person?


A CD player is mounted on four cylindrical rubber blocks. Each cylinder has a height of 0.030 m and a cross-sectional area of 1.2×10–3 m2, and the shear modulus for rubber is 2.6× 106 N/m2. If a horizontal force of magnitude 32 N is applied to the CD player, how far will the unit move sideways? Assume that each block is subjected to one-fourth of the force.

Example 12


A person who weighs 670 N steps onto a spring scale in the bathroom, and the spring compresses by 0.79 cm. (a) What is the spring constant? (b) What is the weight of another person who compresses the spring by 0.34 cm?


The drawing shows a 160-kg crate hanging from the end of a steel bar. The length of the bar is 0.10 m, and its cross-sectional area is 3.2×10–4 m2. Neglect the weight of the bar itself and determine (a) the shear stress on the bar and (b) the vertical deflection DY of the right end of the bar.


 74 *  

In Figure 10.9, the radius of the reference circle is 0.500 m. Suppose the frequency of the simple harmonic motion of the shadow is 2.00 Hz. At time t=0.0500 s, calculate (a) the displacement x, (b) the magnitude of the velocity, and (c) the magnitude of the acceleration of the shadow.

 75 *  

Interactive Solution 10.75 presents a model for solving this problem. A spring (spring constant=112 N/m) is mounted on the floor and is oriented vertically. A 0.400-kg block is placed on top of the spring and pushed down to start it oscillating in simple harmonic motion. The block is not attached to the spring. (a) Obtain the frequency (in Hz) of the motion. (b) Determine the amplitude at which the block will lose contact with the spring.


 76 *  

A block rests on a frictionless horizontal surface and is attached to a spring. When set into simple harmonic motion, the block oscillates back and forth with an angular frequency of 7.0 rad/s. The drawing shows the position of the block when the spring is unstrained. This position is labeled “x=0 m.” The drawing also shows a small bottle located 0.080 m to the right of this position. The block is pulled to the right, stretching the spring by 0.050 m, and is then thrown to the left. In order for the block to knock over the bottle, it must be thrown with a speed exceeding v0. Ignoring the width of the block, find v0.


 77 *  

Consult Interactive Solution 10.77 to explore a model for solving this problem. A spring is compressed by 0.0620 m and is used to launch an object horizontally with a speed of 1.50 m/s. If the object were attached to the spring, at what angular frequency (in rad/s) would it oscillate?


 78 *  

A piece of mohair from an Angora goat has a radius of 31×10–6 m. What is the least number of identical pieces of mohair that should be used to suspend a 75-kg person, so the strain DL/L0 experienced by each piece is less than 0.010? Assume that the tension is the same in all the pieces.

 79 *  

The front spring of a car’s suspension system has a spring constant of 1.50×106 N/m and supports a mass of 215 kg. The wheel has a radius of 0.400 m. The car is traveling on a bumpy road, on which the distance between the bumps is equal to the circumference of the wheel. Due to resonance, the wheel starts to vibrate strongly when the car is traveling at a certain minimum linear speed. What is this speed?


 80 **  

A 0.200-m uniform bar has a mass of 0.750 kg and is released from rest in the vertical position, as the drawing indicates. The spring is initially unstrained and has a spring constant of k= 25.0 N/m. Find the tangential speed with which end A strikes the horizontal surface.


 81 **  

A steel wire is strung between two supports attached to a ceiling. Initially, there is no tension in the wire when it is horizontal. A 96-N picture is then hung from the center of the wire, as the drawing illustrates, so the ends of the wire make angles of 26° with respect to the horizontal. What is the radius of the wire?



 82 **  

A copper rod (length=2.0 m, radius=3.0×10–3 m) hangs down from the ceiling. A 9.0-kg object is attached to the lower end of the rod. The rod acts as a “spring,” and the object oscillates vertically with a small amplitude. Ignoring the rod’s mass, find the frequency f of the simple harmonic motion.

 83 **  

The drawing shows a top view of a frictionless horizontal surface, where there are two springs with particles of mass m1 and m2 attached to them. Each spring has a spring constant of 120 N/m. The particles are pulled to the right and then released from the positions shown in the drawing. How much time passes before the particles are side by side for the first time at x=0 m if (a) m1=m2=3.0 kg and (b) m1=3.0 kg and m2=27 kg?



 84 **  

The drawing shows two crates that are connected by a steel wire that passes over a pulley. The unstretched length of the wire is 1.5 m, and its cross-sectional area is 1.3× 10–5 m2. The pulley is frictionless and massless. When the crates are accelerating, determine the change in length of the wire. Ignore the mass of the wire.


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