C-1Energy Conservation and the Pendulum
1Potential and kinetic energy and the pendulum
a. Relative to the point of equilibrium, where is the energy of the pendulum mostly kinetic energy?
b. Relative to the point of equilibrium, where is the energy of the pendulum mostly potential energy?
2Deriving an equation for predicting maximum speed
a. Determine an equation for predicting the maximum speed of the pendulum by applying the law of conservation of energy. Show your work, and the final equation.
3Testing the equation
Angle(degrees) / Predicted speed
(m/s) / Measured time
(sec) / Measured speed
(m/s)
4Deriving an equation to estimate the period
a. Graph measured speed vs. angle. On the same graph, plot the predicted speed (theoretical) from energy conservation. Draw the theory curve as a solid line and leave the measured points as unconnected dots.
b. How does the theory compare with the measurements?
c. The total distance the pendulum travels in one cycle is approximately 4x, where x is defined in the instruction sheet. Derive an equation for finding total distance (4x) using the length of the string (l) and the starting amplitude (i).
d. Derive an equation to estimate the period by calculating the time it takes the pendulum to travel the distance you calculated above.
5Which factors affect the period of the pendulum?
a. What is the effect of angle on the period of the pendulum? Calculate the part of the equation from 4(c) that depends on the angle. Write this piece of your equation in Table 2 the column headings, between the parenthesis, ( ) . Then, plug in each angle into this piece of the equation. Write your results in the blank spaces of Table 2 below. Note that at 0 degrees, the answer does not work. Use a very small value (0.0001) to estimate the value as a limit approaching zero.
Table 2: Angle and Period
Angle (degrees) / ( ) / Angle (degrees) / ( )0 / 20
5 / 25
10 / 30
15 / 35
b. What do your calculations show?
c. Between 0 and 30 degrees, how much does this term in your equation change? Express your answer as a percent of the value at zero degrees.
d. Record your calculations in Table 3
Table 3: String Length and Period
String Length (m) / Measured Period (sec) / l/ge. Make a graph showing the measured period versus the value of l/g.
f. Use the slope of the graph to obtain a measured value for the numerical coefficient predicted by your theory.
(Hint: square both sides of the equation. Isolate √l/g, and calculate a value for the rest of the factors in the equation)
g. How does this experimentally obtained value of n compare with your approximate theory?