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 = 21.8 N.

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 = 21.8 N.

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)?
with a peg being placed 4/5 of the way down it's moving 2.03 m/s
how do I do this?

Answer 1

To find the tension in the string at the same vertical height as the peg, we can use the principle of conservation of energy. The energy of the system at the top of the pendulum is equal to the energy at the bottom of the pendulum.

At the top of the pendulum, all the energy is potential energy (PE), given by the equation:

PE = m * g * h

Where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the vertical height of the pendulum.

At the bottom of the pendulum, the energy is a combination of kinetic energy (KE) and potential energy:

KE = (1/2) * m * v^2

Where v is the velocity of the mass at the bottom of the pendulum.

Since the pendulum is released from rest, we can state that the initial potential energy equals the final kinetic energy.

m * g * h = (1/2) * m * v^2

We can cancel out the mass (m) from both sides of the equation, giving us:

g * h = (1/2) * v^2

Now we can calculate the height (h) at the bottom of the pendulum when the mass reaches a velocity of v = 2.3 m/s:

h = (1/2) * v^2 / g

h = (1/2) * (2.3^2) / 9.8 = 0.266 m

Now, we know that the peg is placed at 4/5 of the height of the pendulum, so the vertical height directly to the right of the peg will be (1 - 4/5) times the height of the pendulum:

h_right = (1 - (4/5)) * h = (1/5) * h = (1/5) * 0.266 = 0.0532 m

Finally, to find the tension in the string at this height, we can use the equation of potential energy at any given height:

PE = m * g * h_right

To calculate the tension, we need to consider that the weight of the mass is balanced by the tension in the string:

T = m * g

Therefore, the tension in the string at the same vertical height as the peg can be calculated as:

T_right = PE / h_right = (m * g * h_right) / h_right = m * g

So, the tension in the string at the same vertical height as the peg is equal to the weight of the mass, which is T = m * g.