What is the acceleration of a pendulum at its maximum deflection?

In summary, a small sphere with mass m is attached to a weightless string of length 0.5m to form a pendulum, swinging with a maximum angle of 60 degrees with the vertical. To determine its velocity when it passes through the vertical position, the equation 1/2mv^2 = mgh is used, where the vertical height is 0.5m. The resulting velocity is 3.1 m/s. To find the instantaneous acceleration at the maximum deflection, the equation ac = v^2/r is used, with the maximum velocity of 3.1 m/s and a radius of 0.5m, resulting in an acceleration of 6.2 m/s^2.
  • #1
southernbelle
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Homework Statement


A small sphere of mass m is fastened to a weightless string of length 0.5m to form a pendulum. The pendulum is swinging so as to make a maximum angle of 60 degrees with the vertical.
a) What is the velocity of the sphere when it passes through the vertical position?
b) What is the instantaneous acceleration when the pendulum is at its maximum deflection?

Homework Equations





The Attempt at a Solution


a) What is the velocity of the sphere when it passes through the vertical position?
I think that K should be = U because they only change is the pendulum's speed. Is that correct?
1/2mv2= mgh

So the m's cancel and you are left with:
1/2v2 = 9.8(0.43)
v = 2.9 m/s

Did I do that correctly?

b) What is the instantaneous acceleration when the pendulum is at its maximum deflection?
I am not sure where to begin for this part.
 
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  • #2
Hi southernbelle! :smile:
southernbelle said:
a) What is the velocity of the sphere when it passes through the vertical position?
I think that K should be = U because they only change is the pendulum's speed. Is that correct?
1/2mv2= mgh

So the m's cancel and you are left with:
1/2v2 = 9.8(0.43)
v = 2.9 m/s

Did I do that correctly?

Nope … you used sin60º …

but that's the sideways height :cry:

I think Newton would have wanted you to use the vertical height! :wink:
b) What is the instantaneous acceleration when the pendulum is at its maximum deflection?
I am not sure where to begin for this part.

Hint: how does centripetal acceleration depend on speed? :smile:
 
  • #3
The acceleration is usually the derivative of the speed, correct? But I don't think it would be = 0.
 
  • #4
Acceleration is the derivative of displacement, yes, although its relation to linear velocity/speed is given by

[tex]a_c = \frac{v^2}{r}[/tex] which you've hopefully seen before.
 
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  • #5
southernbelle said:
The acceleration is usually the derivative of the speed, correct?
Acceleration is the derivative of the velocity, a vector.

Note that the acceleration has two components: A radial (or centripetal) component and a tangential component. Consider them both.
 
  • #6
Could I use
a= gcos60?

Although I guess I can use the length of the string as r in the equation
ac = v2/r
 
  • #7
southernbelle said:
Could I use
a= gcos60?

Although I guess I can use the length of the string as r in the equation
ac = v2/r

Didn't they ask you for a at the point of maximum deflection? What is its v at that point?
 
  • #8
I got 3.1 for the v.
I did:

1/2mv2 = mgh
v2=2gh
v2 = 2 [9.8(0.5)]
v=3.1m/s

Is that correct? Should I have used 0.5 for the height?

If that is right, then the acceleration would equal 3.1/0.5 = 6.2 m/s2
 
  • #9
southernbelle said:
I got 3.1 for the v.
I did:

1/2mv2 = mgh
v2=2gh
v2 = 2 [9.8(0.5)]
v=3.1m/s

Is that correct? Should I have used 0.5 for the height?

If that is right, then the acceleration would equal 3.1/0.5 = 6.2 m/s2

But what is the height? Wasn't the deflection only 60 degrees?
 
  • #10
Yes, the deflection is 60 degrees.

I used 0.5 as the height because that is the length of the string.
 
  • #11
difference in height

southernbelle said:
Yes, the deflection is 60 degrees.

I used 0.5 as the height because that is the length of the string.

Hi southernbelle! :smile:

The height, h, in the formula PE = mgh,

is the difference in height between the initial and final position.

('cos you're using conservation of energy, so it's the change that matters :wink:)

So in this case it's the vertical amount by which the sphere is higher …

at the bottom position it's at 0.5 below the hinge, and at 60º to the vertical it's at … ? :smile:
 

1. What is a pendulum and how does it work?

A pendulum is a weight attached to a string or rod that is suspended from a fixed point. When the weight is pulled to one side and released, it swings back and forth in a regular pattern. This motion is called oscillation and is caused by the force of gravity acting on the weight.

2. How does the acceleration of a pendulum change with its length?

The length of a pendulum affects its acceleration due to gravity. The longer the pendulum, the slower it will swing and the lower its acceleration will be. This is because the longer pendulum has a larger arc and takes longer to complete one swing.

3. What factors affect the acceleration of a pendulum?

The acceleration of a pendulum is affected by its length, mass, and the force of gravity. The longer the pendulum, the slower the acceleration. The heavier the weight, the faster the acceleration. And the stronger the force of gravity, the faster the acceleration.

4. How does air resistance impact the acceleration of a pendulum?

Air resistance can have a small effect on the acceleration of a pendulum. As the pendulum swings, it pushes against the air molecules in its path, causing some resistance. This resistance can slightly slow down the pendulum's acceleration, but it is often negligible in most cases.

5. Can a pendulum have negative acceleration?

Yes, a pendulum can have negative acceleration. When a pendulum swings back and forth, its acceleration is constantly changing direction. At the highest point of the swing, the acceleration is zero. As it moves downward, the acceleration becomes negative until it reaches the lowest point of the swing and then starts increasing again.

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