A 250g pendulum is suspended 50.cm below a support by a string. The ball is moved sideways along the swing arc so that it is 4.0 cm higher than it is at the bottom of its swing. The ball is then released calculate the speed of the ball at the bottom of the swing
To solve this problem, we can use the principle of conservation of energy.
The potential energy of the pendulum at the lowest point of its swing is given by:
PE_bottom = m * g * h_bottom
Where:
m = mass of the pendulum = 250g = 0.25 kg
g = acceleration due to gravity = 9.8 m/s^2
h_bottom = height of the pendulum at the bottom of its swing = 0
The potential energy of the pendulum when it is 4.0 cm higher than the bottom of its swing is given by:
PE_higher = m * g * h_higher
Where:
h_higher = 4.0 cm = 0.04 m
Since the pendulum is released from rest, all of the potential energy at the higher point is converted to kinetic energy at the lowest point. Therefore:
PE_higher = KE_bottom
KE_bottom = 0.5 * m * v^2
Where:
v = velocity of the pendulum at the bottom of its swing
Setting these two expressions for potential energy equal to each other:
m * g * h_higher = 0.5 * m * v^2
Simplifying:
g * h_higher = 0.5 * v^2
v^2 = 2 * g * h_higher
v = sqrt(2 * g * h_higher)
v = sqrt(2 * 9.8 * 0.04)
v = sqrt(0.784)
v ≈ 0.88 m/s
Therefore, the speed of the ball at the bottom of its swing is approximately 0.88 m/s.