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 calculate the potential energy of the system at different angles, we need to consider the gravitational potential energy equation:
Potential energy = mass * gravity * height
Let's solve each part step by step:
(a) To find the potential energy when the string makes an angle of 45 degrees with the vertical, we need to calculate the height of the mass.
First, we can determine the height using trigonometry. The vertical component of the string can be found using the formula:
Vertical component = string length * cosine(angle)
Given that the string length is 0.95 m and the angle is 45 degrees, we can calculate the vertical component:
Vertical component = 0.95 m * cosine(45°)
Next, we can find the height by subtracting the vertical component from the string length:
Height = string length - vertical component
Now, we can calculate the potential energy:
Potential energy = mass * gravity * height
Substituting the given values:
Potential energy = 0.13 kg * 9.8 m/s^2 * height
(b) To compare the magnitude of the change in potential energy, we need to calculate the potential energy at 0 degrees. As the string becomes vertical, the angle is 0 degrees. We can follow the same steps as in part (a) to calculate the potential energy at this angle.
(c) Similarly, to calculate the potential energy when the string is vertical (angle of 90 degrees), we can follow the steps described above.
Once we have calculated the potential energies at each angle, we can compare their magnitudes to answer part (b).