a pendulum bob with a mass of 0.13 kg is attached to a string with a length of 0.95 m. we choose the potential energy to be zero when the string makes an angle of 90 degrees with the vertical.
(a) find the potential energy of this system when the string makes an angle of 45 degrees with the vertical.
(b) is the magnitude of the change in potential energy from an angle of 90 degrees to 45 degrees greater than, less than, or the same as the magnitude of the change from 45 degrees to 0 degrees? Explain
(c) calculate the potential energy of the system when the string is vertical.
*please list all steps
To solve this problem, we can use the equation for gravitational potential energy.
(a) The potential energy (PE) of an object in a gravitational field is given by the equation PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the vertical height.
First, we need to find the height h when the string makes an angle of 45 degrees with the vertical.
1. Draw a diagram of the pendulum bob at an angle of 45 degrees with the vertical.
```
/
/|
/ | h
/ |
L / |
/ θ |
/ |
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```
2. Using trigonometry, we can determine the height h. We know that the length of the string (L) is 0.95 m and the angle (θ) is 45 degrees.
We can use the sine function to find h:
sin(θ) = h / L
Solving for h, we have:
h = L * sin(θ) = 0.95 m * sin(45 degrees) ≈ 0.6725 m
3. Now we can calculate the potential energy. Remember, the potential energy is zero when the string makes an angle of 90 degrees with the vertical.
PE at 45 degrees = m * g * h
Given:
m = 0.13 kg
g = acceleration due to gravity ≈ 9.8 m/s²
h = 0.6725 m
PE at 45 degrees = 0.13 kg * 9.8 m/s² * 0.6725 m ≈ 0.868 J
Therefore, the potential energy of the system when the string makes an angle of 45 degrees with the vertical is approximately 0.868 Joules.
(b) To compare the magnitude of the change in potential energy from an angle of 90 degrees to 45 degrees with the magnitude of the change from 45 degrees to 0 degrees, we need to calculate both changes.
Change in potential energy from 90 degrees to 45 degrees: PE(45 degrees) - PE(90 degrees)
Change in potential energy from 45 degrees to 0 degrees: PE(0 degrees) - PE(45 degrees)
We can now compare these two changes to determine if they are greater than, less than, or the same.
(c) To calculate the potential energy of the system when the string is vertical (angle of 0 degrees), we can use the same formula as before.
PE at 0 degrees = m * g * h
To find the height h when the string is vertical, we can use the cosine function:
cos(θ) = h / L
But in this case, cos(0 degrees) = 1, so h = L * cos(0) = 0.95 m * 1 = 0.95 m
Now we can calculate the potential energy:
PE at 0 degrees = 0.13 kg * 9.8 m/s² * 0.95 m = 1.1854 J
Therefore, the potential energy of the system when the string is vertical (angle of 0 degrees) is approximately 1.1854 Joules.
To solve this problem, we can use the equation for gravitational potential energy:
Potential energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
Given information:
- Mass of the pendulum bob, m = 0.13 kg
- Length of the string, L = 0.95 m
(a) To find the potential energy of the system when the string makes an angle of 45 degrees with the vertical, we need to determine the height of the bob from its lowest point. We can do this by finding the vertical component of the string length.
Step 1: Calculate the height (h) of the pendulum bob when the string makes an angle of 45 degrees.
h = L * sin(45)
= 0.95 * sin(45)
≈ 0.672 m
Step 2: Calculate the potential energy using the gravitational potential energy equation.
PE = m * g * h
= 0.13 * 9.8 * 0.672
≈ 0.848 J
Therefore, the potential energy of the system when the string makes an angle of 45 degrees with the vertical is approximately 0.848 J.
(b) To determine if the magnitude of the change in potential energy from an angle of 90 degrees to 45 degrees is greater than, less than, or the same as the magnitude of the change from 45 degrees to 0 degrees, we need to compare the heights.
Step 1: Calculate the height (h) when the string makes an angle of 90 degrees.
h_90 = L * sin(90)
= 0.95 * sin(90)
= 0.95 m
Step 2: Calculate the change in potential energy from 90 degrees to 45 degrees.
Change in PE_90to45 = PE_45 - PE_90
= (0.13 * 9.8 * 0.672) - (0.13 * 9.8 * 0.95)
≈ -0.383 J
Step 3: Calculate the change in potential energy from 45 degrees to 0 degrees.
Change in PE_45to0 = PE_0 - PE_45
= (0.13 * 9.8 * 0) - (0.13 * 9.8 * 0.672)
≈ -0.084 J
Comparing the magnitudes of the changes:
Magnitude of change from 90 degrees to 45 degrees (|-0.383|) is greater than the magnitude of change from 45 degrees to 0 degrees (|-0.084|).
Therefore, the magnitude of the change in potential energy from an angle of 90 degrees to 45 degrees is greater than the magnitude of the change from 45 degrees to 0 degrees.
(c) To calculate the potential energy of the system when the string is vertical (angle of 0 degrees), we can use the same gravitational potential energy equation as before.
Step 1: Calculate the height (h) when the string is vertical.
h_0 = L * sin(0)
= 0.95 * sin(0)
= 0
Step 2: Calculate the potential energy when the string is vertical.
PE_0 = m * g * h
= 0.13 * 9.8 * 0
= 0 J
Therefore, the potential energy of the system when the string is vertical (angle of 0 degrees) is 0 J.