Length of a pendulum based on speed and max angle

[lu22]

Homework Statement

The maximum speed of a pendulum is .55 m/s. if the pendulum makes a maximum angle of 8 degrees with the vertical what is the length of the pendulum? The back of the book says that the answer is 1.59m.

Homework Equations

Its in the chapter of our book entitled 'Conservation of Energy' so I'm assuming I use the equations dealing with Kinetic & Potential Energy

The Attempt at a Solution

I tried using 1/2mv^2 = mv^2 / r but i got r = 2m for an answer and the back of the book says its 1.59m so that can't be the right equation

 

[fzero]

I tried using 1/2mv^2 = mv^2 / r

Please examine the units of the LHS of this equation and compare them to the units of the RHS. There's no way that this formula can be correct.

Check your notes and text for a formula for the angular displacement of a pendulum. Try to use that to solve the problem. Post your questions and work if you're still stuck.

 

[lu22]

I'm still stuck, the only equation i could find that had anything to do with an angle was F = ma = mv^2 / r, but that is for uniform circular motion. any more help would be appreciated

 

[fzero]

Ah ok, I didn't notice the part where you said this was in the section on kinetic and potential energy. We're interested in two special points in the motion, namely when the pendulum is at the top of the swing and when the pendulum is at the bottom of the swing.

The bottom of the swing is the lowest point that the pendulum reaches and we measure the potential energy with respect to this point. It's conventional to set the potential energy at the bottom equal to zero. What then is the velocity of the pendulum at the bottom of the swing? What is the kinetic energy at the bottom? What is the total energy at the bottom?

Now, at the top of the swing, what is the velocity of the pendulum? What is the potential energy (relative to the bottom of the swing)? What is the total energy at the top of the swing?

 

[lu22]

Okay, the velocity at the bottom of the swing is given as .55m/s and the velocity at the top is zero. The potential energy at the bottom would be zero and the kinetic energy at the top would be zero. the total energy of the system remains constant so the kinetic energy at the bottom should be equal to the potential energy at the top. I don't know how to find either of those without the mass of the pendulum and I still don't know how to relate this all to the length of the pendulum.

And thanks for all your help!

 

[fzero]

If you can express the kinetic energy at the bottom in terms of the mass, m, and the potential energy at the top in terms of the mass, m, and vertical height, h, then the mass will drop out of the energy conservation law.

To relate h to the length of the pendulum and the maximum angle, draw a diagram of the pendulum and try fitting in right triangles until you find one that works.

 

[lu22]

so if i have 1/2mv²=mgh the mass can cancel out and i will have 1/2v²=gh

would i then just fill in the given velocity and 9.8 for g to solve for the height? or where does the angle come in?

 

[fzero]

so if i have 1/2mv²=mgh the mass can cancel out and i will have 1/2v²=gh

would i then just fill in the given velocity and 9.8 for g to solve for the height? or where does the angle come in?

That equation does let you compute h, which is the vertical height between the top and bottom of the swing. You need a convenient right triangle to relate h to the length of the pendulum.

 

[lu22]

okay now I'm completely stumped. I calculated h to be .028 but I don't know how to line up the right triangles to find the length of the pendulum.

 

[fzero]

okay now I'm completely stumped. I calculated h to be .028 but I don't know how to line up the right triangles to find the length of the pendulum.

Don't forget that h has units. It's a common error, but it's important to keep track of the units for many reasons.

As for the diagram, draw the pendulum at the highest point and extend a vertical line down from the base of the pendulum. There's now two obvious right triangles that you can draw by extending a line from the bottom of the pendulum to the vertical. One of them is easier to use because you can easily figure out the values of two of the sides.

 

[lu22]

Thanks for all your help! I don't have the problem quite down yet but its due tomorrow morning and I can't stay up any later working on it. Thanks again, I definitely understand more than I did when I started!