Conservation of energy in a pendulum

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In the discussion about the conservation of energy in a pendulum, a 2.0 kg bob is given a horizontal velocity of 4.5 m/s, and the analysis focuses on its motion at a 30-degree angle with the vertical. The conservation of energy equation is applied, leading to the conclusion that the initial kinetic energy is not zero, and the gravitational potential energy must be calculated relative to the lowest point. The tension in the string is not simply equal to mg; instead, it requires analyzing the radial components of forces acting on the bob. Additionally, the angle of the string at the bob's highest point can be determined using trigonometric relationships based on the height achieved. Understanding the radial and tangential components of forces is crucial for solving the problem accurately.
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A pendulum consists of a 2.0kg bob attached to a light 3.0m long string. While hanging at rest with the string vertical, the bob is struck a sharp horizontal blow, giving it a horizontal velocity of 4.5m/s. At the instant the spring makes an angle of 30degrees with the vertical.

What is:
a) the speed
b)gravitational potential energy(relative to its value is at the the lowest point)
c)the tension on the string
d)what is the angle of the string with the vertical when the bob reaches its greatest height

The work from the tension of the string is 0 because it is perpendicular to the displacement of the bob.
Ki = 0 since it starts out at rest

The conservation of energy is:
Ui + Ki = Uf + Kf
mgh + 0 = 0 1/2 m Vf^2
vf = root of 2gh

Is this correct approach to answer part A?

For Part B
b)gravitational potential energy(relative to its value is at the the lowest point)
Im not sure what to do. Is this the potential energy of the bob?

c)the tension on the string
Isn't the tension just MG?

d)
d)what is the angle of the string with the vertical when the bob reaches its greatest height
Any hints on how to start this one?
 
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maniacp08 said:
Ki = 0 since it starts out at rest

The conservation of energy is:
Ui + Ki = Uf + Kf
mgh + 0 = 0 1/2 m Vf^2
vf = root of 2gh

Is this correct approach to answer part A?
The initial KE is not zero. The bob starts with a speed of 4.5m/s.

For Part B
b)gravitational potential energy(relative to its value is at the the lowest point)
Im not sure what to do. Is this the potential energy of the bob?
Yes. Find the height of the bob.

c)the tension on the string
Isn't the tension just MG?
No. Analyze the forces acting on the bob at the moment in question.

d)
d)what is the angle of the string with the vertical when the bob reaches its greatest height
Any hints on how to start this one?
What's the final KE in this case?
 
So it is mgh + 1/2 m*(4.5m/s)^2 = 0 + 1/2 m * Vf^2
2gh + (4.5m/s)^2 = Vf^2

The height wouldn't be 3m correct?
Would the height be 3m - 3m * cos 30?

Forces acting on the Bob is
T - mg = m a
Since it is a pendulum, the acceleration of the bob is centripetal correct?
a = V^2/R
where R = the length of the string

D)
The final KE would be 0 since it will stop moving? I am not too sure on w hat you mean.
 
maniacp08 said:
So it is mgh + 1/2 m*(4.5m/s)^2 = 0 + 1/2 m * Vf^2
It starts out at the bottom (h = 0) and ends up at the height h, so:
0 + 1/2 m*(4.5m/s)^2 = mgh + 1/2 m * Vf^2

The height wouldn't be 3m correct?
Would the height be 3m - 3m * cos 30?
Right!

Forces acting on the Bob is
T - mg = m a
The forces on the bob are tension (acting radially) and gravity (acting down).
Since it is a pendulum, the acceleration of the bob is centripetal correct?
a = V^2/R
where R = the length of the string
There are two components to the acceleration: The centripetal (radial) component, which you just described, and a tangential component. To find the tangential component, find the force component in the tangential direction.

D)
The final KE would be 0 since it will stop moving?
Right.
 
What is:
a) the speed
0 + 1/2 m*(4.5m/s)^2 = mgh + 1/2 m * Vf^2
where h = 3m - 3m*cos 30
I get Vf = 3.5m/s

b)gravitational potential energy(relative to its value is at the the lowest point)
MGH = 2kg * 9.81m/s * 3m-3m* cos 30
= 7.9J
c)the tension on the string
So this equation is correct? T - mg = m a

find the force component in the tangential direction.
I have to find the force that made it go horizontal with the speed of 4.5m/s and using that find the horizontal component of it?

d)what is the angle of the string with the vertical when the bob reaches its greatest height
How would I find the angle?
The final KE is = 0
The greatest height would be final potential energy correct? which is mgh.
 
maniacp08 said:
What is:
a) the speed
0 + 1/2 m*(4.5m/s)^2 = mgh + 1/2 m * Vf^2
where h = 3m - 3m*cos 30
I get Vf = 3.5m/s

b)gravitational potential energy(relative to its value is at the the lowest point)
MGH = 2kg * 9.81m/s * 3m-3m* cos 30
= 7.9J
Good.


c)the tension on the string
So this equation is correct? T - mg = m a
No, since tension and mg act in different directions. To find the tension, analyze the radial components of the forces on the bob and apply Newton's 2nd law in the radial direction. Hint: What's the radial component of the weight?


I have to find the force that made it go horizontal with the speed of 4.5m/s and using that find the horizontal component of it?
No. You only have to worry about the forces acting on the bob when it's at 30 degrees.

d)what is the angle of the string with the vertical when the bob reaches its greatest height
How would I find the angle?
The final KE is = 0
The greatest height would be final potential energy correct? which is mgh.
Good. Find the height, then use a bit of trig to find the angle.
 
No, since tension and mg act in different directions. To find the tension, analyze the radial components of the forces on the bob and apply Newton's 2nd law in the radial direction. Hint: What's the radial component of the weight?
Im not too familiar with the radial components and tangential components...,
Im not sure how to calculate them either,
the weight of the bob is mg...For Part D:
finding the height since the KE = 0 and Uf = mgh and Ui = 0
Ki = Uf
1/2 m vi ^2 = mgh
and solve for h?

The h would be the length for the opposite side of the triangle
The hyp would be the length of the string?
Using inverse Sin to find the angle?
 
maniacp08 said:
Im not too familiar with the radial components and tangential components...,
Im not sure how to calculate them either,
the weight of the bob is mg...
Radial just means "along the radius", which in this case means parallel to the string. What's the component of the weight parallel to the string?

For Part D:
finding the height since the KE = 0 and Uf = mgh and Ui = 0
Ki = Uf
1/2 m vi ^2 = mgh
and solve for h?
Yes.

The h would be the length for the opposite side of the triangle
The hyp would be the length of the string?
Using inverse Sin to find the angle?
You know how to find h given the angle, just reverse the process.
 
Radial just means "along the radius", which in this case means parallel to the string. What's the component of the weight parallel to the string?

oooh. So it is mg cos 30?
 
  • #10
maniacp08 said:
So it is mg cos 30?
Yes! But what sign should it have?

I suggest making the tension positive--towards the center of the arc = positive, away = negative. (So what would that make the sign of the centripetal acceleration?)
 
  • #11
The centripetal acc. will be negative?
Im still little confuse here.
So is it T - mg cos 30 = m a where a is the centripetal acc.?
 
  • #12
maniacp08 said:
The centripetal acc. will be negative?
Which way does the centripetal acceleration point?
Im still little confuse here.
So is it T - mg cos 30 = m a where a is the centripetal acc.?
Yep.
 

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