A ball whirls around in a vertical circle at the end of a string. The other end of the string is fixed at the center of the circle. Assume that the total energy of the ball-Earth system remains constant.
(a) What is the tension in the string at the bottom? (Use the following as necessary: m for mass of the ball, g for gravitational acceleration, vb for velocity at the bottom, and R for radius of the circle.)
(b) What is the tension in the string at the top? (Use the following as necessary: m, g, vt for velocity at the top, and R.)
(c) How much greater is the tension at the bottom? (Use the following as necessary: m, g.)
To solve these questions, we can apply the principle of conservation of energy. The total energy of the ball-Earth system remains constant, which means that the sum of the kinetic energy (KE) and potential energy (PE) at any point in the motion is equal to the total energy E.
(a) To find the tension in the string at the bottom, we can use the principle of conservation of energy. At the bottom of the motion, the ball's kinetic energy is at its maximum, and its potential energy is at its minimum. The total energy is given by:
E = KE + PE
Since the ball is at the bottom, the potential energy is zero. The kinetic energy at the bottom can be calculated using the formula:
KE = (1/2) * m * vb^2
where m is the mass of the ball and vb is the velocity of the ball at the bottom.
Therefore, at the bottom of the motion, the total energy is equal to the kinetic energy:
E = (1/2) * m * vb^2
Now, the tension in the string at the bottom is equal to the net force acting on the ball, which is the centripetal force. The centripetal force can be calculated using the formula:
Fc = m * (vb^2 / R)
where R is the radius of the circle.
Since the tension in the string is equal to the centripetal force, the tension at the bottom is:
T_bottom = m * (vb^2 / R)
(b) To find the tension in the string at the top, we can use the same approach. At the top of the motion, the ball's potential energy is at its maximum, and its kinetic energy is at its minimum. The total energy is still equal to the kinetic energy:
E = (1/2) * m * vt^2
where vt is the velocity of the ball at the top.
The centripetal force at the top is still given by:
Fc = m * (vt^2 / R)
Therefore, the tension in the string at the top is:
T_top = m * (vt^2 / R)
(c) To find the difference in tension, we can subtract the tension at the top from the tension at the bottom:
T_difference = T_bottom - T_top
= m * (vb^2 / R) - m * (vt^2 / R)
= m * (vb^2 - vt^2) / R
So, the difference in tension at the bottom is:
T_difference = m * (vb^2 - vt^2) / R
Note that in parts (a), (b), and (c), we assume the ball is not swinging so high that the string becomes slack.