# Pendulum involving conservative energy

• insertnamehere
In summary, the conversation is about finding the maximum height a ball reaches when released from rest and subjected to a constant wind force while connected to a pivot point by a string. The solution involves using the equation FLsin(theta)=mgH and integrating it to find the work done by the wind. The solution also involves finding a new equilibrium position and using the vector sum of mg and F to calculate the new g' value.

#### insertnamehere

Hey! I need help with this question:
A ball having mass m is connected by a strong string of length L to a pivot point and held in place in a vertical position. A wind exerting constant force of magnitude F is blowing from left to right. If the ball is released from rest, show that the maximum height H reached by the ball, as measured from its initial height, is H= (2L)/(1+(mg/F)^2) Check that the above result is valid both for cases when 0<H,L and for L<H<2L.
So far, I know that I'm supposed to use FLsin(theta)=mgH
and integrate it -> W=(integral)FLcos(theta)d(theta)
But I have no idea at all where to go from here! Please, i really need help now! I don't have much time left for this!

insertnamehere said:
Hey! I need help with this question:
A ball having mass m is connected by a strong string of length L to a pivot point and held in place in a vertical position. A wind exerting constant force of magnitude F is blowing from left to right. If the ball is released from rest, show that the maximum height H reached by the ball, as measured from its initial height, is H= (2L)/(1+(mg/F)^2) Check that the above result is valid both for cases when 0<H,L and for L<H<2L.
So far, I know that I'm supposed to use FLsin(theta)=mgH
and integrate it -> W=(integral)FLcos(theta)d(theta)
But I have no idea at all where to go from here! Please, i really need help now! I don't have much time left for this!
The solution to the pendulum motion without the wind is:

$$\theta = \theta_0,sin(\omega t)$$ where $\omega = \sqrt{g/L}$

When you add the wind, the energy added on the forward cycle is lost on the reverse cycle so this is equivalent to a new vibration about a different equilibrium position in which the new g' is determined from the vector sum of mg and F.

$$g' = \sqrt{g^2 +(F/m)^2}$$ and

[tex]\theta_0' [/itex] is the angle that the vector sum of F and mg makes to the vertical. Try working out H from that.

AM

Sure, I can help you with this question. Let's break it down step by step.

First, let's define some variables:
m = mass of the ball
L = length of the string
F = magnitude of the wind force
H = maximum height reached by the ball
g = acceleration due to gravity (9.8 m/s^2)

Now, let's look at the forces acting on the ball. We have the weight force (mg) acting downwards, and the wind force (F) acting horizontally. Since the ball is released from rest, there is no initial velocity and no kinetic energy, so all of the energy is in the form of potential energy. This means we can use the conservation of energy principle, which states that the total energy of a system remains constant.

We can express this principle as:
Initial energy = Final energy

At the initial position, the ball is at rest and has no potential energy, so the initial energy is 0. At the final position, the ball has reached its maximum height H and has potential energy equal to mgh. Therefore, we can write:

0 = mgh

Now, let's look at the work done by the wind force. We know that work is equal to force times distance, so we can write:

Work = F * L

But we also know that work is equal to the change in energy, so we can write:

Work = Final energy - Initial energy

Substituting in our values, we get:

F * L = mgh - 0

Solving for h, we get:

h = FL/mg

Now, we can use this value for h in our initial equation and solve for H, the maximum height reached by the ball:

H = FL/mg + L

But we also know that FLsin(theta) = mgH, so we can substitute this in and solve for H:

H = (FLsin(theta))/mg + L

Since sin(theta) is always less than or equal to 1, we can say that:

H ≤ (FL)/mg + L

Now, let's look at the two cases given in the question:
1. 0 < H, L
2. L < H < 2L

For case 1, since L < H, we can say that (FL)/mg + L < H. Therefore, the maximum height reached by the ball is less than the value we calculated, which

## 1. How does the conservation of energy apply to pendulum motion?

The conservation of energy principle states that energy cannot be created or destroyed, only transformed from one form to another. In the case of a pendulum, the potential energy at the highest point of the swing is converted into kinetic energy at the bottom of the swing and vice versa. This means that the total energy of the pendulum remains constant throughout its motion.

## 2. What is the role of gravity in a pendulum's motion?

Gravity plays a crucial role in the motion of a pendulum. It provides the force that pulls the pendulum back towards its equilibrium position, causing it to oscillate back and forth. Without gravity, the pendulum would not have a restoring force and would not be able to swing.

## 3. How does the length of a pendulum affect its energy?

The length of a pendulum does not affect its energy, but it does affect the period of its motion. According to the law of conservation of energy, the total energy of a pendulum remains constant. However, the longer the pendulum, the longer the period of its oscillation. This means that the pendulum will have a slower swing and will take more time to complete one cycle.

## 4. Can a pendulum's energy be changed or manipulated?

Yes, a pendulum's energy can be changed or manipulated by altering its height or mass. The potential energy of a pendulum is directly proportional to its height, so raising or lowering the pendulum will change its energy. Additionally, adding or removing weight from the pendulum will also affect its energy due to the change in its kinetic energy.

## 5. How does air resistance affect the energy of a pendulum?

Air resistance, also known as drag, can cause the pendulum to lose energy as it swings back and forth. This is because the air molecules collide with the pendulum, creating a resistive force that acts in the opposite direction of its motion. The higher the air resistance, the more energy will be lost and the shorter the pendulum's swing will become.