A pendulum consists of a small object hanging from the ceiling at the end of a string of negligible mass. The string has a length of 0.88 m. With the string hanging vertically, the object is given an initial velocity of 1.7 m/s parallel to the ground and swings upward on a circular arc. Eventually, the object comes to a momentary halt at a point where the string makes an angle θ with its initial vertical orientation and then swings back downward. Find the angle θ.

Question

A pendulum consists of a small object hanging from the ceiling at the end of a string of negligible mass. The string has a length of 0.88 m. With the string hanging vertically, the object is given an initial velocity of 1.7 m/s parallel to the ground and swings upward on a circular arc. Eventually, the object comes to a momentary halt at a point where the string makes an angle θ with its initial vertical orientation and then swings back downward. Find the angle θ.

Answer 1

mv²/2= mgh=mgL(1-cosθ)

cosθ= 1- (v²/gL)
θ=arccos[1- (v²/gL)]

Answer 2

This doesn't give me the correct answer according to the assignment program :/

Answer 3

To find the angle θ, we can use the conservation of energy principle. At the moment when the object comes to a momentary halt, all of its initial kinetic energy will be converted to gravitational potential energy.

First, let's calculate the initial kinetic energy of the object. The formula for kinetic energy is given by KE = 0.5 * m * v^2, where m represents the mass of the object and v represents its initial velocity.

In this case, the mass of the object is not given. However, since the mass does not affect the value of θ, we can ignore it and focus on finding the angle instead.

Using the given values:

Initial velocity, v = 1.7 m/s

Now, we can calculate the initial kinetic energy:

KE = 0.5 * m * v^2 = 0.5 * (1.7)^2 = 0.5 * 2.89 = 1.445 J (Joules)

Next, let's calculate the gravitational potential energy at the moment when the object comes to a halt. The formula for gravitational potential energy is given by PE = m * g * h, where g represents the acceleration due to gravity and h represents the height or vertical displacement.

At the moment when the object comes to a halt, the height is given by h = L * (1 - cosθ), where L represents the length of the string.

Using the given values:

Length of the string, L = 0.88 m

Now, we can calculate the vertical displacement:

h = L * (1 - cosθ) = 0.88 * (1 - cosθ)

Finally, equating the initial kinetic energy to the gravitational potential energy, we have:

KE = PE
1.445 J = m * g * h
1.445 J = m * g * (0.88 * (1 - cosθ))

Since the mass, m, and acceleration due to gravity, g, are constants, we can simplify the equation and solve for cosθ:

cosθ = 1 - (1.445 / (m * g * 0.88))

Please note that we still have an unknown value, the mass of the object, which we can't determine solely from the given information. Thus, we can't provide a specific numerical value for the angle θ. However, you can calculate the angle once you know the mass of the object.