What is the angle at which a ball falls off a hemisphere?

In summary, to find the angle at which a ball falls off a frictionless hemisphere with radius r and starting with initial velocity 0, one can use the equations mgsintheta and mgcostheta to find where the centripetal force is 0. This can also be solved using energy and substituting the calculated velocity into the force equation. This problem can be solved both with and without calculus, by finding the point in which the normal force becomes zero when the ball is constrained to remain on the surface of the hemisphere.
  • #1
bazookajason
9
0

Homework Statement


A ball with initial velocity 0 and mass m starts at the top of a hemisphere and begins to slid down the frictionless side. The radius of the hemisphere is r. At what angle does the ball fall off the hemisphere?

Homework Equations


mgsintheta, mgcostheta, fn
mv^2/r

The Attempt at a Solution


I guess that you can split mg into mgsin theta and mg cos theta and that one of them would create centripetal force. Then you can find where the centripetal force is 0 because there would be no normal force when the ball falls off? I'm not sure. Also, my teacher mentioned that there is a way to solve this with and without calculus.
 
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  • #2
Find the place (or time) at which the normal force becomes zero, if you constrain the ball is constrained to remain on the surface of the hemisphere.
 
  • #3
you can use energy to find theta, solve v then substitute into your force equation
 

FAQ: What is the angle at which a ball falls off a hemisphere?

What is the concept of ball falling of a hemisphere?

The concept of ball falling of a hemisphere refers to the gravitational force acting on a ball placed on the curved surface of a hemisphere. The ball will roll down the surface due to the force of gravity until it reaches the bottom of the hemisphere.

What factors affect the speed of a ball falling off a hemisphere?

The speed of the ball falling off a hemisphere is affected by the height from which it is released, the mass of the ball, and the angle of the hemisphere's curvature. Other factors such as air resistance and surface friction may also play a role.

How does the shape of the hemisphere affect the ball's motion?

The shape of the hemisphere plays a significant role in the ball's motion as it determines the angle of the curved surface and the distance the ball will travel before reaching the bottom. A steeper curvature will result in a faster and shorter fall, while a shallower curvature will result in a slower and longer fall.

What happens if the ball is released from the top of the hemisphere?

If the ball is released from the top of the hemisphere, it will experience a brief moment of weightlessness before being pulled down by the force of gravity. The ball will then roll down the curved surface until it reaches the bottom, where it will come to a stop.

How is the concept of ball falling off a hemisphere applied in real life?

The concept of ball falling off a hemisphere is applied in various fields, including physics, engineering, and sports. It helps in understanding the motion of objects on curved surfaces and designing structures such as roller coasters and water slides. In sports, it is used in the design of tracks and courses for events like bobsledding and luge.

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