# Pendulum Problem: Find Min Release Angle to Clear Peg

• dkoppes
In summary, the ball must be traveling at a minimum velocity of 60 degrees in order for the string not to slack.
dkoppes

## Homework Statement

A pendulum is formed from a small ball of mass m on a string of length L. As the figure shows, a peg is height h = L/3 above the pendulum's lowest point.

From what minimum angle must the pendulum be released in order for the ball to go over the top of the peg without the string going slack?

## The Attempt at a Solution

I attempted to calculate the potential energy that the ball starts with at when it is released using PE = (L-L*cos(theta))*m*g and then using the potential energy at the top of the swing around the peg PE = 2/3*L*m*g and then setting them equal to each other find theta. I also tried a couple different methods that we basically just stabs in the dark and these that I just gave are the only ones that make sense. So far I have tried theta = 48.2, 70.5, and 19.5 but it says that all of these are wrong. Please help me.

#### Attachments

• knight_Figure_10_54.jpg
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Start by asking yourself how fast must the ball be going at the top of its motion to keep the string taut. (Hint: Use Newton's 2nd law.)

I did try that, but I still couldn't figure it out, that was one of the methods that I tried.

Show us what you exactly tried, with respect to Doc Al's advice.

first, find out the tension of the rope at the top as a function of angle. What is the relationship between the tension and gravity? what is the net force? What happens when the tension is zero?

for the rope to not have any slack in it, doesn't tension have to = m*g

At what point are you considering?

So once you have centripetal acceleration, find the minimum velocity at the top of the swing. At the top of the swing, how much how much potential energy does the ball have? How much higher does the ball have to start out? You were on the right track before, but you have to factor in the minimum kinetic energy, its not zero

Remember that you have to account for the kinetic AND the potential energy of the ball after it hits the peg and swings around.

So...

PE = KE + PE

Im also having trouble with this problem.

I first solved for the critical velocity (when the force normal = 0) and got sqrt(g*L/3).
I then used the conservation of energy equation,
1/2(m)(vf)^2+(m)(g)(yf)=1/2(m)(vi)^2+(m)(g)(yi)
Since vi = 0, and the masses cancel, I was left with
1/2(vf)^2+(m)(g)(yf) =(g)(yi)

I then solved for v:

(vf)^2=2(gyf-gyi)

so

(L/3)g = 2(L-Lcos(theta))-2(L/3)

substituting the critical velocity for vf and factoring out the L and g, I solved for theta and got 60 degrees.

Can anyone please see where I'm going wrong?

Thanks!

never mind, wrong equation! thanks

I am having trouble with this one. Which equation was wrong?

## 1. What is a pendulum problem?

A pendulum problem involves finding the minimum release angle at which a pendulum can swing in order to clear a peg or obstacle.

## 2. Why is finding the minimum release angle important?

The minimum release angle determines the minimum amount of energy needed for the pendulum to successfully clear the peg. This is important for optimizing the design and efficiency of pendulum-based systems.

## 3. How is the minimum release angle calculated?

The minimum release angle can be calculated using the principles of conservation of energy and conservation of momentum. It involves finding the angle at which the pendulum's kinetic energy is equal to the potential energy at the point of release.

## 4. What factors affect the minimum release angle?

The length and mass of the pendulum, as well as the height and position of the peg, all affect the minimum release angle. Other factors such as air resistance and friction may also play a role.

## 5. How is the pendulum problem relevant in real-world applications?

The pendulum problem has real-world applications in engineering and physics, such as in designing pendulum-based clock mechanisms and amusement park rides. It also has applications in sports, such as determining the optimal release angle for a golf swing or a baseball pitch.

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