Question

A ball of mass m is attached to a string of length L. It is being swung in a vertical circle with enough speed so that the string remains taut throughout the ball's motion. Assume that the ball travels freely in this vertical circle with negligible loss of total mechanical energy. At the top and bottom of the vertical circle, the ball's speeds are v_t and v_b, and the corresponding tensions in the string are T_t and T_b. T_t and T_b (vectors) have magnitudes T_t and T_b.

Find T_b - T_t, the difference between the magnitude of the tension in the string at the bottom relative to that at the top of the circle.
Express the difference in tension in terms of m and g. The quantities v_t and v_b should not appear in your final answer.

Answer 1

The difference between the tensions in the string at the top and bottom of the vertical circle can be expressed in terms of the mass and acceleration due to gravity.

Step-by-step explanation:

When a ball of mass m is being swung in a vertical circle with a taut string of length L, the tensions in the string at the top and bottom of the circle are T_t and T_b respectively. At the top of the circle, the ball is moving at its slowest speed, so the tension T_t in the string is negligible. At the bottom of the circle, the ball is moving at a faster speed, and the tension T_b in the string is greater than T_t.

The difference between the magnitudes of the tensions at the bottom and top of the circle, T_b - T_t, can be expressed in terms of the mass m and acceleration due to gravity g as:

T_b - T_t = m(g + v_t^2/L - v_b^2/L)

where v_t and v_b are the speeds of the ball at the top and bottom of the circle respectively.

Answer 2

The difference in the magnitude of tension in the string at the bottom relative to the top in a vertical circular motion is 5m*g. This result is obtained by applying the conservation of mechanical energy and the requirement for centripetal force at both the top and bottom of the motion.

Step-by-step explanation:

To find the difference in the tension of a string (Tb - Tt) at the bottom and top of a vertical circular motion, we need to apply principles from mechanics. At the top, the tension Tt and the weight of the ball (m*g) both act downward. The centripetal force needed to keep the ball in circular motion is provided by the net force from both tension and weight: Tt + m*g = m*vt2/L. At the bottom, the tension Tb acts upwards, while weight acts downwards. The centripetal force here is provided solely by the tension, which has to overcome the weight: Tb - m*g = m*vb2/L.

Since the mechanical energy is conserved, the kinetic energy difference equals the potential energy difference: m*g*2*L = 0.5*m*(vb2 - vt2). From this, we find vb2 = vt2 + 4*g*L.

The difference in tension is Tb - Tt = m*(vb2/L - g) - (m*vt2/L + m*g). By substituting vb2 and simplifying, we get Tb - Tt = 5m*g.