A mass hangs on the end of a massless rope. The pendulum is held horizontal and released from rest. When the mass reaches the bottom of its path it is moving at a speed v = 2.3 m/s and the tension in the rope is T = 22.4 N. Now a peg is placed 4/5 of the way down the pendulum’s path so that when the mass falls to its vertical position it hits and wraps around the peg. Return to the original mass. What is the tension in the string at the same vertical height as the peg (directly to the right of the peg)?

Question

A mass hangs on the end of a massless rope. The pendulum is held horizontal and released from rest. When the mass reaches the bottom of its path it is moving at a speed v = 2.3 m/s and the tension in the rope is T = 22.4 N. Now a peg is placed 4/5 of the way down the pendulum’s path so that when the mass falls to its vertical position it hits and wraps around the peg. Return to the original mass. What is the tension in the string at the same vertical height as the peg (directly to the right of the peg)?

Answer 1

To find the tension in the string at the same vertical height as the peg (directly to the right of the peg), we can use the principle of conservation of energy.

First, let's determine the potential energy (PE) and kinetic energy (KE) of the mass at the bottom of its path.

Given:
Speed of the mass at the bottom (v) = 2.3 m/s
Tension in the rope at the bottom (T) = 22.4 N

The potential energy at the bottom is zero since it is at the lowest point of the pendulum's swing.

The kinetic energy at the bottom is given by the formula:
KE = (1/2)mv^2,

where m is the mass of the object and v is the velocity.
Since the mass is cancelled out in the equation, we have:
KE = (1/2)v^2.

So, the kinetic energy at the bottom is:
KE_bottom = (1/2)(2.3)^2 = 2.645 J.

Now, let's consider the situation when the mass is at the same vertical height as the peg (directly to the right of the peg). At this position, the mass has the potential energy due to its height.

Using the principle of conservation of energy, we can equate the initial kinetic energy at the bottom (KE_bottom) to the potential energy at this height (PE_h) plus the new kinetic energy (KE_new), which is given by:

KE_bottom = PE_h + KE_new.

Since the mass is at the same height as the starting position, the potential energy is zero, and we get:

2.645 J = 0 + KE_new,
KE_new = 2.645 J.

Finally, we can find the new velocity (v_new) at this height using the formula for kinetic energy:

KE_new = (1/2)m(v_new)^2.

Rearranging the equation, we have:

(v_new)^2 = 2KE_new/m,
(v_new)^2 = 2(2.645)/m.

Now, here comes the twist in the problem. We need to return to the original mass and its tension. Unfortunately, we are not given the mass. Without the mass, we cannot determine the new velocity (v_new) and, subsequently, the tension in the string at the same vertical height as the peg.

Therefore, we need additional information or clarify the problem further to proceed with the calculation.