Object rolling off a sphere

In summary, the problem involves a small object sliding down a frictionless solid sphere, losing contact with the sphere at one point. The angle between a line drawn from this point to the center of the sphere and the horizontal can be found using the formula v^2/R = gsin(theta) and the Conservation of Energy equation. The angle is approximately 41.8 degrees.
  • #1
mistymoon_38
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0
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?
 
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  • #2
Whenever something loses contact with a surface the normal force vanishes.
 
  • #3


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.
 

1. How does the shape of the sphere affect the object rolling off?

The shape of the sphere can greatly affect the path of the object rolling off. If the sphere is perfectly round, the object will roll off in a straight line. However, if the sphere is irregularly shaped or has bumps or grooves, the object's path may be altered.

2. Does the weight of the object rolling off matter?

Yes, the weight of the object can affect the speed at which it rolls off the sphere. Heavier objects will have more momentum and may roll off at a faster rate than lighter objects.

3. What factors determine the distance the object will roll off the sphere?

The distance the object rolls off the sphere is determined by the initial velocity of the object, the angle at which it rolls off, and the friction between the object and the sphere's surface.

4. How does the surface of the sphere affect the object rolling off?

The surface of the sphere can affect the object's rolling by providing more or less friction. A rougher surface will slow the object down and possibly alter its path, while a smooth surface will allow the object to roll more easily.

5. Can the object roll off a sphere at an angle?

Yes, the object can roll off the sphere at an angle depending on the initial velocity and the angle at which it is released. The angle at which the object rolls off can also affect the distance it travels and its path.

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