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Physics 107

Review - Simple Harmonic Motion


Imagine that you video tape 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) = A cos 2 πt/T        (Equation 1)
where A 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.


Fig. 1b below 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(A sin 2 πt/T)      (Equation 2)
Find (a) from Eq. 2 an expression for the maximum velocity in symbols and (b) from Fig. 1b the magnitude of the maximum velocity.


Fig. 1c below 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 πT)2 (A cos 2 πt/T)      (Equation 3)
Find (a) from Eq. 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.


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


For x(t) = A cos 2 πt/T,  xo = A when  t = 0.
Since v(t) = -2 π/T(A sin 2 πt/T),  vo = 0.

In other words for this equation, the initial position xo equals the maximum displacement A from the equilibrium position and the initial velocity vo = 0. You can provide for xo ≠ A and vo ≠ 0 by writing:
x(t) = A cos (2 πt/T+ δ)                 (Equation 1')
where δ is the phase constant that depends on xo and vo. Show that Equation 1' guarantees the definition of (a) the period T and (b) the amplitude A. (c) Find xo and vo  when δ = - π/2 and describe where the mass is initially and how it is moving.


A massless spring with constant k exerts a force on a mass m when the spring is extended a distance x. By Newton’s second law of motion, -kx = ma. Since the acceleration a = dv/dt = d/dt(dx/dt) = d2x/dt2,

-kx = m d2x/dt2      or            d2x/dt2 + (k/m)x = 0            (Equation 1")

Show that                              x(t) = A cos (2 πt/T + δ)            (Equation 1')

is a solution to Eq. 1" if T = 2 π(m/k)1/2.  Hint: find d2x/dt2 and substitute it into Eq. 1'.


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


Given that x(t) = A cos (ωt + δ) find (a) x(0) = xo and (b) v(0) = vo.  Find in terms of xo,  vo,  and ω = 2πν = 2 πT (c) tan δ and (d) A.


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 xo.  (a) Find xo 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 + xo), and (c) find the frequency of the motion when the spring is released.


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.


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


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.


Figure 4 below represents a particle of mass m moving with uniform circular motion. Find the following components of r and a in the X-direction : (a) displacement x and (b) the acceleration ax in terms of r and a function of Θ = ωt. (c) Find the ratio of the acceleration -ax to the displacement x. (d) Given that ω = 2 πf, where f is the frequency of the motion, express the ratio found (c) in terms of ω. (e) Is the projection of uniform circular motion on the X-axis a case of simple harmonic motion? (f) Use your results from (d) and Problem #4 to express the frequency f, the number of oscillations per second, in terms of the mass m and spring constant k.  Repeat the above for the components of r and a in the Y-direction.


The x coordinate of a particle obeys the equation b2 d2x/dt2 + c2x = 0, where b and c are constants. What is the period of oscillation?


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.


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.


A mass-spring system has frequency of 3/2 s-1.  Find A and δ for
x(t) = A cos( πt + δ) if xo = 0.25 m and vo = -1.5 m/s.


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.


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?


We wish to find the fraction of the mass ms of a spring that contributes to the load when a system is vibrating given that the kinetic energy of an object of mass m and velocity v is given by K = 1/2 mv2.  For Fig. 6 below, find (a) the differential mass dm of the spring of length dy, (b) the velocity of the element assuming that the velocity varies linearly from v at y = L to 0 at y = 0, (c) the kinetic energy of the element, and (d) the total kinetic energy of the spring. Your answer will be in the form of 1/2( )ms v2.  The fraction in the ( ) is your answer.


Fig. 7a below is a plot of the extension of a spring as a function of the force exerted on it.  Fig. 7b below is a plot of the square of the period of a mass-spring system as a function of the mass attached to the spring. The spring has a mass ms and an effective mass ms’.  (a) Find the spring constant k from Fig. 7a. (b) Write an expression for the period of a spring with an effective mass ms’ as a function of the added mass m. (c) From the intercept on the T2-axis of Fig. 7b, find ms’  and then find ms.  (d) There is another way to find ms’.
Hint: What is m when T2 = 0?  Show that this method gives the same value of ms’  as that in (b).


A flat plate P moves in simple harmonic motion with a frequency of 1.5 s-1.  A block rests on the plate (Fig. 8 below). The coefficient of static friction between the block and the plate is 0.60. Find the magnitude of the amplitude of the motion if the block does not slip on the plate.


Most rigid bodies in the everyday world are at rest and thus in stable equilibrium. For a body in stable equilibrium, the net force acting on it is zero. If a body is displaced a distance x away from the position of stable equilibrium the force F will be of opposite sign to x.  In general the force is of the form
F = c1 x + c2x2 + c3x3 + . . . .,
where c1 is a negative constant.  The slope of F vs x at x = 0 is (dF/dx) o = c1. For small oscillations about the origin, F = ma becomes
c1x = m d2x/dt2 or d2x/dt2 + (-c1/m)x = 0.
Comparing this with the "standard" equation for simple harmonic motion,
d2x/dt2 + (k/m)x = 0,
we see that the "equivalent" spring constant k is
-c1 = -(dF/dx)o = -[d(-dU/dx)/dx]o = (d2U/dx2)o.
Suppose the potential energy in a diatomic molecule can be expressed as the sum of an attractive term b/r3 and a shorter-range repulsive term a/r5, that is, U = (a/r5) - (b/r3).  Find (a) the equilibrium position ro,  (b) the "spring constant k"  between the two atoms, and (c) the vibrational frequency of this diatomic molecule if each atom has a mass m.


An ice cube is given a small displacement from the bottom of a spherical bowl of radius R. Given the equation of the circle in Fig. 21 below is x2 + (R - y)2 = R2, find (a) the potential energy U in terms of m, g, x, and R,  (b) -dU/dx,
(c) Fx for x << R,  and (d) the period of oscillation when it is released, assuming no friction.


Given that the potential energy of a mass-spring system is 1/2 kx2 show that (a) the total energy of the system E = 1/2 kA2 = U + K = 1/2 kx2 + 1/2 mv2 and (b) v = [k/m(A2 - x2)]1/2 = dx/dt. (c) Separate the variables x and t in (b). Integrate this expression (limits on t from 0 to t and limits on x from xo to x) to show that x(t) = A sin [(k/m)1/2t + δ]. The initial phase angle δ = sin -1 (xo/A).


A pendulum consists of a rod of mass m1 and a point mass m2 (Fig. 23 below). Find (a) the torque and moment of inertia about the pivot point,  (b) d2Θdt2 in terms of the torque and the moment of inertia and (c) the period for small oscillations in terms of m1,  m2,  L, and g.


A thin uniform disk with a radius R swings about an axis that consists of a thin pin driven through the disk, as shown in Fig. 24 below.  Find (a) the moment of inertia and torque about the axis, (b) d2Θ/dt2 in terms of the moment of inertia and the torque, (c) the period of oscillation and (d) the value of r for a minimum period of oscillation and the value of the period for this r.

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Susan D. Kunk
Phyllis J. Fleming
October 8, 2002
October 29, 2002