How Does Circular Motion Impact Calculations in Physics Problems?

[jibjab]

Hi everyone, I am taking my first physics class ever and I'm lost! I have a couple of questions that are giving me a lot of trouble. Any assistance you can give me is very much appreciated!

Homework Statement

A turntable rotates counterclockwise at 85 rpm. A speck of dust on the turntable is at (theta)0.31 rad at t=0 s. What is the angle of the speck at t=7.77 s? Answer should be between 0 and 2(pi) rad.

The Attempt at a Solution

For this, I changed 85 rpms to 1.4 rps. I tried to find the angular velocity by dividing 2(pi)rad/1.42 rps, and my answer was 4.42 rad/s. Then I plugged the info into the equation (theta final)=0.31 + (4.42 rad/s)(7.77 s) and got 34.65.
I divided that answer by 2(pi).
I got 5.51 X 2(pi) rad, following the example in the book. I ended up with a final answer of 183 deg, which was wrong.

Homework Statement

The passengers in a roller coaster car feel 58% heavier than their true weight as the car goes through a dip with a 36.1 m radius of curvature. What is the car's speed at the bottom of the dip?

The Attempt at a Solution

I really don't understand how you figure this out without their true weight?

Homework Statement

An Earth satellite moves in a circular orbit at a speed of 5247.66 m/s. What is its orbital period?

The Attempt at a Solution

I used the equation T=2(pi)X(square root)r^3/GM <-- (I hope that makes sense)
using G=6.67 X 10^-11, M=5.98 X 10^24 and r=6.37 X 10^24
My answer was 5057.96 s. I tried a couple other ways but they were wrong too.

Homework Statement

A 182.4 kg block on a 43.1 cm long string swings in a circle on a horizontal, frictionless table at 55 rpm. What is the speed of the block? The tension on the string?

The Attempt at a Solution

This one I just need an equation for, the ones that I've found need either one or the other so I don't know how to go about it?

Thanks in advance for anything you reply to!

 

[Delphi51]

For this, I changed 85 rpms to 1.4 rps. I tried to find the angular velocity by dividing 2(pi)rad/1.42 rps, and my answer was 4.42 rad/s.

Shouldn't that be 2(pi)*1.42 rps?

The rest of your calc looks good to me.

 

[tomcue]

Hi there, I can definitely help you with your questions on circular motion and gravity!

For the first question, you are on the right track with converting the rpm to rps and using the equation (theta final) = (theta initial) + (angular velocity)(time). However, there is a small mistake in your calculation. The angular velocity should be 1.42 rad/s, not 4.42 rad/s. This is because 85 rpm is equivalent to 85/60 = 1.42 rps, not 1.4 rps. So your final answer should be (0.31 rad) + (1.42 rad/s)(7.77 s) = 11.02 rad. Since the answer should be between 0 and 2(pi) rad, we need to convert 11.02 rad to its equivalent in that range, which is 11.02 rad - 2(pi) rad = 1.74 rad.

For the second question, you can actually use the concept of centripetal force to solve for the speed of the roller coaster car. The equation for centripetal force is F = mv^2/r, where F is the centripetal force, m is the mass of the object, v is the speed of the object, and r is the radius of the circular motion. In this case, the centripetal force is equal to the weight of the passengers multiplied by the 58% increase (1.58). So we have F = (1.58mg). Setting this equal to the equation for centripetal force, we have (1.58mg) = mv^2/r. We can rearrange this to solve for v, which gives us v = sqrt((1.58g)r). Plugging in the values given, we get v = sqrt((1.58)(9.8 m/s^2)(36.1 m)) = 15.9 m/s.

For the third question, you are using the correct equation, T = 2(pi)sqrt(r^3/GM). The only mistake is that you have the radius of the Earth (6.37 X 10^24 m) instead of the radius of the orbit (which is the distance from the center of the Earth to the satellite). This distance is equal to the sum of the Earth's radius (6.37 X 10^6 m) and the height