A ball rolls forwards and backwards on a curved track as shown in Fig. 1.1.
ball
flexible track
Fig. 1.1
It is suggested that the period T of the oscillations is related to the radius r of the ball and the
radius of curvature C of the track by the relationship
T 2 = 28π2
5g (C – r )
where g is the acceleration of free fall.
You are provided with a flexible track. Design a laboratory experiment to test the relationship
between T and r. Explain how your results could be used to determine a value for C. You
should draw a diagram, on page 3, showing the arrangement of your equipment. In your
account you should pay particular attention to
(a) the procedure to be followed,
(b) the measurements to be taken,
(c) the control of variables,
(d) the analysis of the data,
(e) the safety precautions to be taken.
let x = C-r
height of center of ball = x(1 - cos Theta)
for small Theta
1 - cos Theta = 1-1+theta^2/2 =theta^2/2
Potential energy = m g x(theta^2)/2
ke of ball about top point = (1/2)mx^2 (d theta/dt)^2
ke of ball around its center
= (1/2) I omega^2
I =(2/5) m r^2
omega= (x/r)d theta/dt
ke = (1/2)(2/5)mr^2 (x^2/r^2) (d theta/dt)^2
theta = A sin w t
d theta/dt = A w cos wt
max Potential = max kinetic
m g x(A^2)/2= A^2w^2 (1/5)m x^2
w^2 = g(5/2x)
w = 2 pi f = 2 pi/T
w^2 =(2 pi)^2/T^2
T^2 = (2 pi)^2 /w^2
= (2 pi)^2 [2(C-r)]/5g
Left out ke about pin at top
let x = C-r
height of center of ball = x(1 - cos Theta)
for small Theta
1 - cos Theta = 1-1+theta^2/2 =theta^2/2
Potential energy = m g x(theta^2)/2
ke of ball about top point = (1/2)mx^2 (d theta/dt)^2
ke of ball around its center
= (1/2) I omega^2
I =(2/5) m r^2
omega= (x/r)d theta/dt
ke = (1/2)(2/5)mr^2 (x^2/r^2) (d theta/dt)^2
theta = A sin w t
d theta/dt = A w cos wt
max Potential = max kinetic
m g x(A^2)/2= A^2w^2 (1/5)m x^2+(1/2)m x^2 A^2 w^2
g /2x = w^2 (1/5+1/2) = w^2(7/10)
w^2 = g(5/7x)
w = 2 pi f = 2 pi/T
w^2 =(2 pi)^2/T^2
T^2 = (2 pi)^2 /w^2
= (2 pi)^2 [7(C-r)]/5g
= 28 pi^2 (C-r)/5g
To design a laboratory experiment to test the relationship between T and r, and determine the value of C, here are the steps to follow:
(a) Procedure:
1. Set up a flexible track on a flat surface, ensuring it is securely anchored at both ends.
2. Attach a ball to one end of the track, making sure it can roll freely.
3. Measure the distance from the ball's starting position to the center of curvature (C) of the track.
4. Adjust the position of the ball so that it rests at a point on the track where the radius of curvature is equal to the measured distance (r).
5. Release the ball from this position and observe its motion, paying attention to the time it takes to complete one full oscillation (T).
6. Repeat steps 3-5 for different radius (r) values, ensuring the ball starts at the center of curvature (C) position each time.
(b) Measurements:
1. Measure the distance from the starting position of the ball to the center of curvature (r).
2. Use a stopwatch to measure the time taken for the ball to complete one full oscillation (T).
(c) Control of variables:
1. Keep the mass of the ball constant throughout the experiment.
2. Keep the angle of the track constant.
3. Make sure the track remains flexible and doesn't change shape during the experiment.
(d) Analysis of the data:
1. Plot a graph with T squared (T^2) on the y-axis and (C - r) on the x-axis.
2. Use the equation T^2 = (28π^2 / 5g) * (C - r) to calculate the theoretical values of T^2.
3. Compare the experimental data with the theoretical values to determine the relationship between T and r and validate the given relationship.
4. Use the relationship to find the value of C by calculating the gradient of the graph.
(e) Safety precautions:
1. Ensure the flexible track is secure and stable to prevent accidents or injuries.
2. Handle the ball with care to prevent it from hitting anyone or causing damage.
3. Keep the track clean and free from any obstacles to ensure smooth ball movement.
4. Follow standard laboratory safety guidelines and procedures.
Please note that for accuracy and reliability, it is essential to conduct multiple trials and record the average values of T and r for each trial.
To design a laboratory experiment to test the relationship between T and r, and determine a value for C, you can follow the following steps:
(a) Procedure to be followed:
1. Set up the flexible track on a flat surface.
2. Attach a small ball to one end of the track.
3. Position the track on an incline so that the ball can roll forwards and backward on the curved track.
4. Ensure that the track is secure and stable.
5. Mark the starting position of the ball at the highest point of the track.
6. Release the ball from rest at this position and measure the time it takes for the ball to complete one full oscillation (forward and backward).
7. Repeat steps 5 and 6 for different values of the radius of the ball (r).
8. Ensure that at least three different radii values are used to obtain a range of data.
(b) Measurements to be taken:
1. Measure the radius of the ball (r) using a ruler or calipers.
2. Measure the time (T) taken for the ball to complete one oscillation using a stopwatch or a timer.
(c) Control of variables:
1. Keep the radius of the track curvature (C) constant throughout the experiment.
2. Keep the starting position of the ball and the angle of inclination of the track constant.
3. Ensure that the track curvature does not change during the experiment.
(d) Analysis of data:
1. Plot a graph with the period squared (T^2) on the y-axis and (C - r) on the x-axis.
2. Perform a linear regression analysis on the data points to determine the relationship between T^2 and (C - r).
3. The slope of the line obtained from the regression analysis will give you a value for 5g.
(e) Safety precautions:
1. Make sure the track is securely set up to prevent it from moving or falling during the experiment.
2. Handle the ball and the track with care to avoid any injuries.
3. Ensure that there is enough space around the experimental setup to prevent any accidents.
By following these steps and analyzing the data, you will be able to test the relationship between T and r and determine a value for C using the provided equation. Remember to record your observations and measurements accurately to obtain reliable results.