A ball on the end of a string is moving in circular motion as a conical pendulum as in the �figure. The length L of the string is 1.70 m, the angle � is 37.0 degrees and the ball completes one revolution every 2.51 seconds.
(a) What is the speed (m/s) of the ball?
(b) What is the acceleration (m/s2 ) of the ball?
To find the speed and acceleration of the ball, we can use the formulae for circular motion.
(a) Speed (v) of the ball:
The speed of an object in circular motion is given by the formula v = 2πr / T, where r is the radius (length of the string) and T is the time taken to complete one revolution.
Given:
Length of the string (L) = 1.70 m
Time taken to complete one revolution (T) = 2.51 s
To find the radius (r), we can use trigonometry. Since the ball is moving in a conical pendulum, the radius (r) is the length of the string projected onto the horizontal plane. It can be calculated using the formula r = L sin(θ), where θ is the angle between the string and the vertical axis.
Given:
Angle (θ) = 37.0 degrees
Length of the string (L) = 1.70 m
Using these values, we can calculate the radius (r):
r = L * sin(θ)
= 1.70 m * sin(37.0 degrees)
= 1.70 m * 0.6018
≈ 1.022 m
Now, we can calculate the speed (v) of the ball using the formula:
v = 2πr / T
= 2 * 3.1416 * 1.022 m / 2.51 s
≈ 4.08 m/s
Therefore, the speed of the ball is approximately 4.08 m/s.
(b) Acceleration (a) of the ball:
The acceleration of an object in circular motion is given by the formula a = v^2 / r.
Using the speed (v) that we found in part (a) and the radius (r), we can calculate the acceleration (a) of the ball:
a = v^2 / r
= (4.08 m/s)^2 / 1.022 m
≈ 16.66 m/s^2
Therefore, the acceleration of the ball is approximately 16.66 m/s^2.