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 find the potential energy of a pendulum bob at different angles, we can use the formula for gravitational potential energy.
(a) The formula for gravitational potential energy is given by:
Potential Energy = mass * gravity * height
Given data:
Mass of the bob (m) = 0.13 kg
Length of the string (L) = 0.95 m
To find the height (h), we need to consider the vertical component of the string when it makes an angle of 45 degrees with the vertical. This can be calculated using trigonometry:
h = L * sin(45)
Since gravitational potential energy is taken as zero at 90 degrees, the change in potential energy is given by:
Change in Potential Energy = mass * gravity * (height at 45 degrees - height at 90 degrees)
Plugging in the values:
Change in Potential Energy = 0.13 kg * 9.8 m/s^2 * (0.95 m * sin(45) - 0.95 m)
Evaluating the expression will give us the change in potential energy.
(b) To determine whether the magnitude of the change in potential energy from 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 values of the two changes. We can calculate the change in potential energy at 0 degrees using the same formula as before, but with sin(0) = 0. The comparison will depend on the numerical values of the two expressions found in step (a).
(c) To calculate the potential energy of the system when the string is vertical (at 0 degrees), we need to calculate the height (h) at that angle. Since the string is vertical, the height will be the same as the length of the string:
h = L
Then we can use the formula:
Potential Energy = mass * gravity * height
Plugging in the values:
Potential Energy = 0.13 kg * 9.8 m/s^2 * 0.95 m
Evaluating the expression will give us the potential energy of the system.