Vertical Circle Rod: Work and Energy Calculations | Simple Homework Question

[Puchinita5]

Homework Statement

Figure 8-29 shows a thin rod, of length L and negligible mass,that can pivot about one end to rotate in a vertical circle. A heavyball of mass m is attached to the other end. The rod is pulled asidethrough an angle 6 and released.As the ball descends to its lowestpoint, (a) how much work doesthe gravitational force do on itand (b) what is the change in thegravitational potential energy ofthe ball-Earth system? (c) If thegravitational potential energy istaken to be zero at the lowestpoint, what is its value just as theball is released? (d) Do the mag-nitudes of the answers to (a)through (c) increase, decrease, orremain the same if angle 6 is in-creased?

Homework Equations

http://www.niu.edu/~debrown/Spring2009/Q4_5.09.pdf if you click this link, scroll down to problem two for the image.

The Attempt at a Solution

i know how to do the problem, the only thing i cannot figure out is how to find the height. I googled the problem and someone said that the change in y should be L-Lcos(theta)...and when i used that it worked, but i don't know why. It's problably some simple geometry thing but I need to know what L-Lcos(theta) means in case i get a problem similar to this on an exam.

 

[rl.bhat]

Draw an right angled triangle containing point of suspension, mass and the vertical line. If l is the length of the rod, its projection on the vertical line is L*cosθ. The remaining length on the vertical position of the rod is L - L*cosθ. That is the height through the mass m rises.

 

[nlightner]

I would like to clarify some concepts and provide a more detailed response to this content.

Firstly, the homework problem is asking for the work and energy calculations for a vertical circle rod, which is a common example used to illustrate the conservation of energy principle. This principle states that energy cannot be created or destroyed, but it can be transferred from one form to another.

In this case, the work done by the gravitational force is equal to the change in gravitational potential energy of the ball-Earth system. The work done by a force is defined as the product of the force and the displacement in the direction of the force. In this problem, the gravitational force is always acting downwards, in the direction of the displacement of the ball.

Now, let's address the concept of height in this problem. The height in this problem refers to the vertical distance between the lowest point of the ball and the starting point of the ball. This height is equal to the length of the rod, L, multiplied by the sine of the angle 6. This can be derived using simple geometry, as the rod and the vertical line form a right triangle with the angle 6 as one of its angles.

Therefore, the change in gravitational potential energy of the ball-Earth system is equal to the product of the mass of the ball, the acceleration due to gravity, and the change in height. This can also be written as mgh, where h is the change in height.

If we take the gravitational potential energy to be zero at the lowest point, then the initial gravitational potential energy is equal to mgh, where h is the initial height of the ball. This initial height can be calculated using the same formula as above, but with the initial angle 6.

Finally, to answer part (d) of the problem, the magnitudes of the work done and the change in gravitational potential energy will remain the same if the angle 6 is increased, as the height and the displacement of the ball will also increase by the same factor.

In conclusion, the concept of height in this problem refers to the vertical distance between the lowest point of the ball and the starting point of the ball, and it can be calculated using simple geometry. The work done by the gravitational force is equal to the change in gravitational potential energy, and this potential energy can be calculated using the mass of the ball, the acceleration due to gravity, and the change in height. I hope this explanation helps to clarify the