potential energy and energy conservation

Question

The pendulum shown to the left consists of a bob of mass of 0.31 kg, attached to a string of length L = 1 meters. It is moving to the right with a speed of 4.33 m/sec at point A. What we'd like to know is what the the velocity of the bob when it reaches point B which is 34 degrees from vertical.

There are two different ways to look at this problem, each of which will give us the same answer. Let's look at each separately. (For all of the questions below, let's start by defining the potential energy of the bob to be zero at point A.)

First, lets look at the question in terms of the work energy theorem. (W_TOT=ΔKE)

1)

What is the initial kinetic energy of the bob at point A?

KE_A =

2)What is the work done by on the bob by gravity as it moves from point A to point B?

W_GRAV =

3)

What is the work done by on the bob by the force of tension from the string attached to the bob as it moves from point A to point B?

W_TEN =

4)

What is the total work done on the bob as it moves from point A to point B?

W_TOT =

5) What is change in kinetic energy of the bob as it moves from point A to point B?

ΔKE = J 

6)

What is kinetic energy of the bob at point B?

KE_B = J 

7)

What is velocity of the bob at point B?

v_B = m/sec 

8)

Now, lets look at the problem in terms of mechanical energy. That is,ME=U+KE and ΔME=W_non-con. (Recall we are defining the potential energy to be zero at point A.) What is the kinetic energy of the bob at point A?

KE_A =J 

9)

What is the potential energy of the bob at point A?

U_A= J 

10)

What is the mechanical energy of the bob at point A?

ME_TOT-A=  J 

11)

What is work done by non-conservative forces as the bob moves from point A to point B?

W_non-con= J 

12)

What is the mechanical energy of the bob at point B?

ME_TOT-B= J 

13)

What is the potential energy of the bob when it reaches point B?

U_B = J 

14)

What is the kinetic energy of the bob when it reaches point B?

KE_B = J 

15)

What is the speed of the bob at point B?

v=

Answer 1

I can't see the image, so I will assume point A is when the bob is at the lowest point of the swing.

1) The kinetic energy of the bob is based on the speed it has at point A. So:

KE = 1/2 mv² = 1/2 (0.31 kg)(4.33 m/s)²

KE_A = 2.9 J

2) The work done by gravity is equal to the bob's gravitational force (its weight) multiplied by its displacement in the vertical direction (a.k.a. change in height). So:

W_GRAV = mgh = (0.31 kg)(9.81 m/s²)(???) <= wait, what's the height?

To get height, we need to look at the picture. From point A to point B, the bob has moved upward a certain distance vertically. Doing some trigonometry, we can see that the amount of vertical distance should be the length of the string (1 meter) minus the vertical distance between the bob at B and the pivot of the pendulum. That's the adjacent side of the triangle, and the hypothenuse is the length of the string, so we use cosine: cosθ = A / H => cos(34°) = A / (1 m) => A = 0.83 m. Hard to explain without a picture!

This means the vertical distance between the bob and the top of the pendulum is 0.83 m. The remaining distance (from the bob to the lowest point A) is 0.17 m, so that's the height of the bob.

W_GRAV = mgh = (0.31 kg)(9.81 m/s²)(0.17 m)

W_GRAV = 0.52 J

3) The work due to the tension force is zero. This is because the tension force and the displacement of the bob are always 90 degrees to each other. If force and displacement are at a right angle, no work is done.

W_TEN = 0

4) The total work is equal to the work done by all forces. Since work by tension is zero, total work is equal to the work done by gravity, or 0.52 J.

W_TOT = 0.52 J

5) According to the work-energy theorem, the change in kinetic energy is equal to the total work done.

ΔKE = W_TOT (we got this in #4)

ΔKE = 0.52 J

6) The original kinetic energy was 2.9 J from #1, and the change in kinetic energy is 0.52 J. But is the change an increase or a decrease? Think about the pendulum. As it goes from A to B, it must slow down, as it gains height. Therefore:

KE_B = KE_A - ΔKE = (2.9 J) - (0.52 J)

KE_B = 2.38 J

7) Using the formula from #1 backwards:

KE = 1/2 mv² => 2.38 J = 1/2 (0.31 kg)v²

v_B = 0.82 m/s

8) We did this before, see #1.

KE_A = 2.9 J

9) Since the bob is at the lowest point, and we defined that point to have zero U:

U_A = 0

10) Total energy is KE + U:

ME_TOT-A = 2.9 J

11) There are no non-conservative forces involved. These would be things like friction or air resistance, and neither are mentioned, so you can consider them negligible.

W_NON-CON = 0

12) No non-conservative forces, so the total mechanical energy will be the same:

ME_TOT-B = 2.9 J (it's just broken up into KE and U now.)

13) We can calculate potential energy with U = mgh.

U_B = (0.31 kg) (9.81 m/s²) (0.17 m) (hey, didn't we do this in #2?)

U_B = 0.52 J

14) The kinetic energy is the same calculation as before:

KE_B = 2.38 J

15) And same for velocity:

v = 0.82 m/s