## Ch 13 Oscillatory Motion

### Questions 2, 4, 6, 8, 9 Additional problems from Serway's fourth edition

(4 ed) 13.1 A 1.5 kg block at rest on a tabletop is attached to a horizontal spring having a spring constant of 19.6 N/m. The spring is initially unstretched. A constant 20.0-N horizontal force is applied to the object causing the spring to stretch.

(a) Determine the speed of the block after it has moved 0.30 m from equilibrium if the surface between the block and the tabletop is frictionless.

(b) Answer part (a) if the coefficient of kinetic friction between block and tabletop is 0.20.

(4 ed) 13.2 A 50-g mass connected to a spring of force constant 35 N/m oscillates on a horizontal, frictionless surface with anamplitude of 4.0 cm. Find

(a) the total energy of the system and

(b) the speed of the mass when the displacement is 1.0 cm.

When the displacement is 3.0 cm, find

(c) the kinetic energy and

(d) the potential energy.

(4 ed) 13.3 A car with bad shock absorbers bounces up and down with a period of 1.5 s after hitting a bump. The car has a mass of 1500 kg and is supported by four springs of equal force constant k. Determine a value for k.

(4 ed) 13.4 A mass m is oscillating freely on a vertical spring (Fig P13.56). When m = 0.810 kg, the period is 0.910 s. An unknown mass on the same spring has a period of 1.16 s. Determine (a) the spring constant k and (b) the unknown mass. Conceptual Questions

Q13.If the coordinate of a particle varies as x = - A cos t, what is the phase constant in Equation 13.3? At what position does the particle begin its motion (for t = 0)?

Equation 13.3, p 391, is

x = A cos( t + )

We have x = - A cos wt

and that is really just

x = A cos (wt + )

so the phase angle or phase constant is (measured in radians). If you prefer it measured in angles then it is 180o.

For t = 0, this gives

x(t=0) = A cos(0 + ) = A cos( ) = A (- 1) = - A

x(t = 0) = - A

Q13.4 Determine whether the following quantities can be in the same direction for a simple harmonic oscillator:

(a) displacement and velocity,

(b) velocity and acceleration,

(c) displacement and acceleration.

Let's begin with Equation 13.3, from page 391,

x = A cos( t + )

v = dx/dt = - A sin( t + )

a = dv/dt = d2x/dt2 = - A 2 cos( t + )

Now we can look at the directions.

Displacement and velocity will sometimes be in the same direction and sometimes in opposite directions.

Velocity and acceleration will sometimes be in the same direction and sometimes in opposite directions.

However, displacement x and accelertion a will always be in opposite directions!

Q13.6 Describe qualitatively the motion of a mass-spring system when the mass of the spring is not neglected.

The spring moves and that involves kinetic energy and it involves F = ma. The moving spring includes extra mass. Including the mass of the springs means we need to describe the mass-spring SHO (simple harmonic oscillator) with a larger mass than just the mass attached to the spring. It turns out that increasing the mass of the system by one-third the mass of the spring is just the needed correction.

Q13.8 A block-spring system undergoes simple harmonic motion with an amplitude A. Does the total energy change if the mass is doubled but the amplitude is not changed? Do the kinetic and potential energies depend on the mass?

The total energy is equal to the maximum potential energy

Etot= Umax = (1/2) k xmax2 = (1/2) k A2

so the total energy does not depend upon the mass. What will the change in mass affect? The maximum kinetic energy is still equal to the total energy,

Etot= Kmax = (1/2) m vmax2

If the mass is increased, then the maximum speed, vmax, must decrease. So changing the mass changes the maximum speed. But the maximum kinetic energy, the maximum potential energy, and the total energy depend only upon the amplitude (and the spring constant k).

Q13.9 What happens to the period of a simple pendulum if the pendulum's length is doubled? What happens to the period if the mass of the bob is doubled?

Lengthening the string of a pendulum increases its period.

The period of a simple pendulum is independent of the mass. Problems from the current (5th) edition of Serway and Beichner.

13.1 The displacement of a particle at t = 0.25 s is given by the expression

x = (4.0 m) cos (3.0 p t + p), where x is in meters and t is in seconds. Determine

(a) the frequency and period of the motion,

(b) the amplitude of the motion,

(c) the phase constant, and

(d) the displacement of the particle at t = 0.25 s 13.3 A 20-g particle moves in simple harmonic motion with a frequency of 3.0 oscillations/s (3.0 Hz) and an amplitude of 5.0 cm.

(a) Through what total distance does the particle move during one cycle of its motion?

(b) What is its maximum speed? Where does that occur?

(c) Find the maximum acceleration of the particle. Where does that occur? 13.12 A block of unknown mass is attached to a spring of spring constant 6.50 N/m and undergoes simple harmonic motion with an amplitude of 10.0 cm. When the mass is halfway between its equilibrium position and the endpoint, its speedis measured to be + 30.0 cm/s. Calculate

(a) the mass of the block,

(b) the period of the motion, and

(c) the maximum acceleration of the block.  13.7 A spring stretches by 3.9 cm when a 10-g mass is hung from it. If a 25-g mass attached to this spring oscillates in simple harmonic motion, calculate the period of motion 13.8 A simple harmonic oscillator takes 12.0 s to undergo five complete vibrations. Find

(a) the period of its motion,

(b) the frequency in Hz, and

(c) the angular frequency in rad/s 13.30 A simple pendulum is 5.0 m long.

(a) What is the period of simple harmonic motion for this pendulum if it is located in an elevator accelerating upward at 5.0 m/s2?

(b) What is the answer to part (a) if the elevator is accelerating down ward at 5.0 m/s2?

(c) What is the period of simple harmonic motion for this pendulum if it is placed in a truck that is accelerating horizontally at 5.0 m/s2?  13.36 A torsional pendulum is formed by attaching a wire to the center of a meter stick with mass 2.0 kg. If the resulting period is 3.0 miutes, what is the torsion constant for the wire? 13.71 A mass m is connected to two springs of force constants k1 and k2 as in Figure P13.71. In each case, the mass moves on a frictionless table and is displaced from equilibrium and released. Show that in each case the mass exhibits simple harmonic with periods of     Solutions to the additional problems from Serway's fourth edition

(4 ed) 13.1 A 1.5 kg block at rest on a tabletop is attached to a horizontal spring having a spring constant of 19.6 N/m. The spring is initially unstretched. A constant 20.0-N horizontal force is applied to the object causing the spring to stretch.

(a) Determine the speed of the block after it has moved 0.30 m from equilibrium if the surface between the block and the tabletop is frictionless.

(b) Answer part (a) if the coefficient of kinetic friction between block and tabletop is 0.20. (4 ed) 13.2 A 50-g mass connected to a spring of force constant 35 N/m oscillates on a horizontal, frictionless surface with anamplitude of 4.0 cm. Find

(a) the total energy of the system and

(b) the speed of the mass when the displacement is 1.0 cm.

When the displacement is 3.0 cm, find

(c) the kinetic energy and

(d) the potential energy. (4 ed) 13.3 A car with bad shock absorbers bounces up and down with a period of 1.5 s after hitting a bump. The car has a mass of 1500 kg and is supported by four springs of equal force constant k. Determine a value for k. (4 ed) 13.4 A mass m is oscillating freely on a vertical spring (Fig P13.56). When m = 0.810 kg, the period is 0.910 s. An unknown mass on the same spring has a period of 1.16 s. Determine (a) the spring constant k and (b) the unknown mass. 