Graviational potentioal energy

[stanton]

1. Homework Statement

A 40 N toy is placed in a light swing that is attached to ropes 2.0m long. Find the gravitational potential energy associated with the toy relative to its lowest position when (a) the ropes are horizontal (b) the ropes make a 30 degrees angle with the vertical and (c) at the bottom of the circular arc.

2. Homework Equations

PEg = mgy (note: y is distance)
Wg = -mg(yf-yi)=mgyi-mgyf=PEi-PEf
Wg=-deltaPE

3. The Attempt at a Solution

I do not know what to start. Well, as my prop show a similar problem with (b) problem, I tried to draw free body diagram. like vertical = y-axis and horizontal = x axis. So the the rope and y-axis makes a sort of triangle. and the unknown length of the one side of triangle must be length of cord multiplied by cos30. And I cant' go further. Maybe I went down all wrong. Will you help?
If I am asked solve the length between the maximum point reached by toy or something like that, I think I can do it. However, about energy, I am confused.

 

[rl.bhat]

At the bottom PE = ...?
a) The rope is horizontal. So the displacement is L ( length of the rope). So PE = ...?
b) The displacement is (L - Lcasθ). So PE =...?

 

[tomcue]

I would like to clarify that gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It is calculated by multiplying the object's mass, the acceleration due to gravity, and the object's height.

In this problem, we can use the equation PEg = mgy to calculate the gravitational potential energy associated with the toy at each position. To start, we need to determine the height of the toy relative to its lowest position at each scenario.

(a) When the ropes are horizontal, the height of the toy is 2.0m. Therefore, the gravitational potential energy would be PEg = (40 N)(9.8 m/s^2)(2.0 m) = 784 J.

(b) When the ropes make a 30 degree angle with the vertical, the height of the toy can be calculated using trigonometry. The height would be 2.0m x cos30 = 1.73m. Thus, the gravitational potential energy would be PEg = (40 N)(9.8 m/s^2)(1.73 m) = 690.4 J.

(c) At the bottom of the circular arc, the height of the toy is 0. Therefore, the gravitational potential energy would be PEg = (40 N)(9.8 m/s^2)(0) = 0 J.

It is important to note that gravitational potential energy is a scalar quantity, meaning it only has magnitude and no direction. Therefore, it is not affected by the angle of the ropes or the direction of the motion of the toy. I hope this helps to clarify the concept of gravitational potential energy and how it relates to this problem.