Energy Conservation in Two Dimensions
Published on Jul 09, 2023
Energy conservation is a cornerstone idea in all of science. In this experiment, we will verify this basic principle by observing the collision of two objects who are moving in a two dimensional plane.
In this experiment, we will,
1. understand the concept of energy conservation,
2. create and observe a fully-translational collision,
3. quantitatively demonstrate the near-elasticity of a collision,
4. learn how to compare theoretical predictions with experimental observations and
5. run through a complete cycle of experiment, data generation, analysis and presentation.
The horizontal distance the struck ball will travel as it lands on the ground is given by
d = 2√ hL(1 cos θ)
where h is the height of the collision point from the platform, L is the length of pendulum and
θ the angle from the vertical above which the swinging balls pendulum is positioned. These
quantities are illustrated in Figure (1c)
The apparatus consists of a ball attached with a thin stainless rod connected with a bearing to form a relatively low-loss pendulum . By construction, the pendulum ball follows a circular arc and is not allowed to twist as it swings through the arc. The pendulum ball is held at its highest position by a thread strung over a pulley like attachment and is shown in Figure (1b). To start the ball in motion either a match can be used to sever the thread  or the thread can be pulled and released manually by hand from the desired position.
Figure 1: (a) The Experimental Setup; (b) labelled side view of the pendulum and struck ball; (c) illustration of quantities:
The length of the pendulum L; height of collision point h; angle from the vertical ; horizontal distance from plumb ball d In order to achieve pure translational motion of the projectile ball, the balls must touch at the exact bottom of the swinging ball's arc in such a way that the line connecting the center of the two balls at collision is in both the plane of the arc and in a horizontal plane. To achieve this precision, a screw adjusts the height of fulcrum to ensure that struck ball leaves the tee along an initially horizontal path. The ball ies o the setup and drops to the oor on a platform where its position is recorded when it strikes a piece of carbon paper, leaving a small dot on the graph paper.
The parameters that are measured before the experiment is performed are L and h. Variables measured during the experiment include , the angle swinging ball pendulum is positioned at, and d, the horizontal distance the struck ball travels before hitting the oor. The linear distances are measured with a meter-stick and the angle is determined from the protractor attached to the apparatus. The horizontal distance of the ball will be determined experimentally and compared with the theoretical prediction.
1. Measure the horizontal distance of ball using meter-stick after performing the experiment.
2. Calculate the percentage error between the predicted and measured distances. F Q 4. Discuss reasons to why there is a di erence between the predicted and measured distances.
3. You are required to present your data graphically. Choose your axes and variables. Plot the uncertainties as well. You will be given points for
(a) selecting the most suitable variables for plotting,
(b) calculating uncertainties in the variables,
(c) comparing a best t to your data and the predicted curves.
4. Note that Q5 is open-ended and you are required to explore and come up with the best strategy of presenting your data. F Q 6. Compare a best t of your data with a graph for the predicted outcome.
5. Use these two graphs to estimate the energy lost in a collision.
 A thin rod with a bearing was used because attempts to construct the setup using threads was not successful. With threads, after the collision the pendulum ball almost always twists back and forth as the two threads on either side of the ball oscillate out-of-plane. This twisting added an intolerable energy loss.
 The thread and match are used in starting the pendulum to ensure that there is no additional momentum to disturb the ball as it begins its swing. This technique was pioneered by Foucault as described in Amir D. Aczel's biography "Pendulum: Leon Foucault and the Triumph of Science" (Atria Books, New York, 2003)