Review - Simple Harmonic Motion

1.

Imagine that you videotape the motion of a mass attached to a spring and measure the displacement x from the equilibrium position as a function of time t. When you plot x versus t, you get a graph resembling Fig. 1a below. The mathematical expression for the curve of Fig. 1a is:

x(t) = x_m cos 2 πt/T   (Equation #1)

where x_m is the amplitude of the motion, the maximum displacement from the equilibrium position and T is the period, the time for one complete oscillation. From Fig. 1a, find the amplitude and the period of the motion.

2.

Fig. 1b above is a plot of the velocity v as a function of the time t. You find v from the slope dx/dt of x versus t.

v(t) = (-2 π/T)(x_m sin 2 πt/T)   (Equation 2)

Find (a) from Equation 2, an expression for the maximum velocity in symbols, and (b) from Fig. 1b, the magnitude of the maximum velocity.

3.

Fig. 1c above is a plot of acceleration a as a function of the time t. You find a from the slope dv/dt of v versus t.

a(t) = -(2 π/T )² (x_m cos 2 πt/T)   (Equation 3)

Find (a) from Equation 3, (i) an expression for the maximum acceleration, and (ii) -a(t)/x(t) in symbols, and (b) from Fig. 1c the magnitude of the maximum acceleration.

4.

When the force on an object is directly proportional to, and in the opposite direction of, the displacement, the motion of the object is simple harmonic.
(a) Show that the motion of a mass attached to a stretched spring is simple harmonic. (b) Find the ratio, -acceleration/displacement.

5.

Equate the expression found for -a/x in Part b of Problem 4 to that found in Problem 3a(ii) to find the period T of the motion for the mass-spring system.

6.

For x(t) = x_m cos 2 πt/T, x_o = x_m when t = 0. v(t) = -2 π/T(x_m sin 2 πt/T), v_o = 0 when t = 0. In other words for this equation, the initial position x_o equals the maximum displacement x_m from the equilibrium position and the initial velocity v_o = 0. You can provide for x_o = x_m and v_o = 0 by writing x(t) =
A cos (2 πt/τ+ φ), (Equation #1), where φ is the phase constant that depends on x_o and v_o.  Show that Eq. 1 guarantees the definition of (a) the period T and (b) the amplitude x_m.  (c) Find x_o and v_o when φ = -π/2 and describe where the mass is initially and how it is moving.

7.

Given x(t) = x_m cos (2 πt/T + φ). Sketch x(t) for φ equal to (a) 0, (b) - π/2, and
(c) - π.  Sketch v(t) for φ equal to (d) 0, (e) - π/2, and (f) - π.  (g) What is the usefulness of φ ?

8.

When a 0.50 kg-object is attached to a vertically supported spring, it stretches 0.10 m from the equilibrium position. Find k for the spring.

9.

A massless spring of constant k is hung vertically and not extended. A mass m is attached to the spring and it stretches a distance x_o.  (a) Find x_o in terms of k, m and g. Now the spring is pulled down an additional distance x.  (b) Write Newton’s second law for total extension (x + x_o), and (c) find the frequency of the motion when the spring is released.

10.

A pendulum bob of mass m attached to a string of length L vibrates back and forth along a circular arc. (a) Draw an isolation diagram for the bob and show all the forces acting on it. (b) Write Newton's second law of motion for the component of the force tangent to the path. (c) For small angles, the horizontal displacement x is approximately equal to the displacement along the arc s (Fig. 2 below). Show that for small displacements, the pendulum vibrates with simple harmonic motion. Find the frequency of its motion.

11.

Given x(t) = x_m cos 2 πt/T for a spring with constant k and attached mass m sketch as a function of t, (a) the displacement x, (b) the velocity v, (c) the acceleration, (d) the potential energy, (e) the kinetic energy, and (f) the total energy.

12.

An object with mass m = 0.60 kg attached to a spring with k = 10 N/m vibrates back and forth along a horizontal frictionless surface. If the amplitude of the motion is 0.050 m, what is the velocity of the object when it is 0.010 m from the equilibrium position.

13.

Given x(t) = 0.01 m cos (0.02 πs^-1 t - π/2). Find (a) the amplitude, (b) the period, (c) the frequency, and (d) the initial phase of the motion.

14.

A particle is executing simple harmonic motion. The displacement x as a function of time t is shown in Fig. 3 below. Find (a) the period, (b) amplitude,
(c) equation of motion, (d) maximum velocity and (e) maximum acceleration.

15.

A pendulum bob swings a total distance of 4.0 cm from end to end and reaches a speed of 10 cm/s at the midpoint. Find the period of oscillation.

16.

The motion of a particle is given by x(t) = 4.0 cm cos ( πt s^-1 - π/6).  Find the velocity of the particle when x = 2.0 cm.

17.

A pendulum consists of a rod of mass m₁ and a point mass m₂ (Fig. 4 below). Find (a) the torque and moment of inertia about the pivot point,  (b) d²Θ/dt² in terms of τ and I and (c) the period for small oscillations in terms of m₁, m₂, L, and g.

18

In Fig. 5a below, two springs of spring constants k₁ and k₂ are connected in parallel. In Fig. 5b below, they are connected in series. Find the frequencies of the systems for (a) Fig. 5a and (b) Fig. 5b.

19.

A massless spring with spring constant k = 10.0 N/m is attached to an object of mass m = 0.300 kg. One third of the spring is cut off. What is the frequency of the oscillations when the "new" spring-mass is set into motion?

20.

Figure 6 below is a plot of the potential energy of a mass-spring system. The total mechanical energy E of the system = 0.200 J. Find (a) the potential energy U and (b) the kinetic energy K at x = 0.025 m. Find (c) the spring constant k and (d) the speed of the particle when x = 0.025 given that the mass of the object m = 0.30 kg. Find (e) the amplitude of the motion and (f) the maximum velocity of the object.