1) Some of the graphs shown below represent the motion of a pendulum -- a massive ball attached to a rigid, nearly massless rod, which in turn is attached to a rigid, nearly frictionless pivot. 

On the right below are shown four graphs of the pendulum's angle as a function of time. On the left are a set of four initial conditions and a denial. Match each graph with its most likely initial conditions (or with the denial). Note that the scales on the y axes are not necessarily the same. (There is not necessarily a one-to-one match.) 

(1) q0 = 120o w0 = 0  (2) q0 = 173o w0 = 30o/s  (3) q0 = 6o w0 = 0  (4) q0 = 173o w0 = 0  (5) Not a possible pendulum graph.

(A)  (B)   (C)  (D)

2) A pendulum consisting of a massive ball on a light but nearly rigid rod is shown at three successive times in the pictures on the below. The maximum angle of displacement of the pendulum from its equilibrium point is 90^o.
 

Copy each of the pictures. On each picture draw three arrows indicating the direction of 

_ the mass's velocity (a single arrow like this: ->; 
_ the total force the mass feels (a double arrow like this: =>; and 
_ the acceleration of the mass (a boldface arrow like this: ).

3) This problem uses the M.U.P.P.E.T. program Pendulum*.

The pendulum satisfies the equation of motion derived in class   

(assuming no damping). We will refer to this as the exact equation. It has to be solved numerically. (This is what the program Pendulum does.) For small angles we can approximate sinq by q. This gives the equation for simple harmonic motion   

which we can solve analytically. We refer to this as the approximate equation. 

(a) In the program Pendulum, set the starting angle to 15^o , the starting velocity and the damping to 0. Look at the curves for the position, velocity, and acceleration of the bob. Do they resemble each other? Do they look like simple harmonic motion? Why? 
(b) Increase the starting angle to 180^o in steps of 15^o. (Do not actually use 180^o. It will crash the program. Use 179^o.) At each step, read the period from the graph. Try to get two significant figures. Plot the period as a function of starting angle. (It might be easier to read if you start with x₀ = 0 and v₀ > 0.) 
(c) Describe what happens to the velocity curve as the angle approaches 180^o. At what angle does it begin to differ visibly from the harmonic oscillator curves? 
(d) Qualitatively compare what the "exact" and "approximate" results look like for the initial conditions 

q(t=0) = 3.0 rad,  rad/sec;

q(t=0) = 6.0 rad,  rad/sec. 

Describe what is happening physically in the two cases. Why is there a qualitative difference between the results in the two methods?

4) Consider a pendulum of mass m and length L. 

(a) Derive the energy conservation equation for the motion of the pendulum. (Do not use the small amplitude approximation.) 
(b) If the pendulum is released from a starting angle of q₀, what will be the maximum speed it travels at any point on its swing? 

(c) When on vacation this summer, I visited a forest which had many waterfalls. At one of the waterfalls, a long rope hung down from the top of the cliff near the waterfall and had a seat on the bottom. Adventurous visitors could hop onto the seat and swing down into the waterfall as shown in the photo at the right. I estimated that their starting angle was about 20o. I also timed their swing and discovered it took them 8 seconds to swing out and back. Estimate the length of the rope and the speed with which they passed through the waterfall. (The top of the rope is not shown in the photo.)

5) A small metal ball of mass m hangs from a pivot by a rigid, light metal rod of length R as shown in the figure on the right. The ball is swinging back and forth with an amplitude that remains small throughout its motion (). Ignore all damping. 

(a) The equation of motion of this ideal pendulum can be derived in a variety of ways and is  . 

For small angles, show how this can be replaced by an approximate equation of motion that can be solved more easily than the one given. 
(b) Write a general solution for the approximate equation of motion you obtained above that works for any starting angle and angular velocity (as long as the angles stay in the range where the approximation is OK). Demonstrate that what you have written is a solution and show that at a time t=0 your solution can have any given starting position and velocity. 
(c) If the length of the rod is 0.3 m, the mass of the ball is 0.2 kg, and the clock is started at a time when the ball is passing through the center (q = 0) and is moving with an angular speed of 0.1 rad/s, find the maximum angle your solution says the ball will reach. Does the approximate equation of motion remain valid for this motion?