What Is the Angle When an Object Loses Contact with a Sphere?

[mistymoon_38]

Object rolling off a sphere...

Homework Statement

A small object begins at the top of a frictionless solid sphere. Its initial speed is negligibly small. The sphere is stationary at all times. The small object then slides down the surface of the sphere. At one point the small object loses contact with the sphere. Draw a line from this point to the center of the sphere. What is the angle between this line and the horizontal?

Homework Equations

The Attempt at a Solution

I have no idea how to get this started. Any hints on where to go with this?

 

[DavidWhitbeck]

Whenever something loses contact with a surface the normal force vanishes.

 

[MysterioIII]

Hey,

I bet you're in my recitation unless our professors somehow took the same problem! Although this problem requires the use of energy, we need to understand that the centripetal force is equal to the gravitational force at the point when the object leaves the sphere.

This gives us v^2/R = gsin(theta). Solve for the velocity here.

Then you want to use the Conservation of Energy equation. You know that the initial potential energy is mgR (we assume the potential energy line is a horizontal line through the center of the sphere). Then when you solve for "Mechanical Energy Final", we know that the potential energy is smaller (mgRsin(theta)) and the kinetic energy is 1/2mv^2. I recommend you solve for v in the energy equation and then solve the two v's against each other.

Through this process, you should find the angle. I got 41.8 degrees, let me know if you disagree.