At what angle does an object fall off a sphere?

In summary, a mass m is placed on the top of a smooth hemisphere of radius a and given a small impulse, causing it to slide down the hemisphere under the influence of gravitational acceleration g. It is found that the mass flies off the surface of the hemisphere when its vertical height has decreased by a/3, or when theta = 48 degrees. The three forces acting on the mass are centripetal acceleration towards the center, mgcos(theta), and mgsin(theta). By using energy considerations, the centripetal acceleration can be found and used to show that N=0 when theta = 48 degrees.
  • #1
Inkage
2
0

Homework Statement


A mass m is placed on the top of a smooth hemisphere of radius a such that theta = pi/2 it is given a very small impulse and as a result begins to slide down one side of the hemisphere under the influence of the gravitational acceleration g.

Show that the mass flies off the surface of the hemisphere when its vertical height has decreased by a/3.

Homework Equations


None

The Attempt at a Solution



I managed to work out that when the vertical height is reduced by a/3, theta = 48 degrees. But I have no idea how to show that the mass flies off the hemisphere. I suppose when N = 0 the object will fall off, but surely that would be when cos(theta) = 0? Which would mean... theta = 90 degrees... Help please >_> I feel a bit retarded for not being able to do this question =(
 
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  • #2
As long as the mass is on the hemisphere (This is circular motion!), what are the forces acting on it? (There are 3)
Find these three forces as dependent on the angle relative to the vertical, [tex]\theta[/tex] and find for what angle [tex]\theta[/tex] it holds that [tex]N=0[/tex]

If you're familiar with accelerated reference frames you should move your observer to the rotating frame of reference.
 
  • #3
The three forces would be... (I think)

1. Centripetal acceleration (towards centre)
2. mg [mgcos(theta)]
3. Normal contact force? mgsin(theta)?

Im so confused =.=
 
  • #4
Inkage said:
The three forces would be... (I think)

1. Centripetal acceleration (towards centre)
2. mg [mgcos(theta)]
3. Normal contact force? mgsin(theta)?

Im so confused =.=

Check your trig. :) Unless you were measuring your [tex]\theta[/tex] from the horizontal, in which case you are correct!

Now use energy considerations to find the centripetal acceleration and you're good to go.
[tex]F_{centripetal} = \frac{mv^2}{R}[/tex]
 

1. What is the angle at which an object would fall off a sphere?

The angle at which an object would fall off a sphere is known as the critical angle. This angle is dependent on the radius of the sphere and the gravitational force acting on the object. It can be calculated using the formula: θ = tan-1(r/g), where θ is the critical angle, r is the radius of the sphere, and g is the gravitational force.

2. Is the angle at which an object falls off a sphere the same for all objects?

No, the angle at which an object would fall off a sphere is not the same for all objects. It depends on the weight, size, and shape of the object, as well as the parameters of the sphere, such as its radius and gravitational force.

3. Can an object fall off a sphere at any angle?

No, an object cannot fall off a sphere at any angle. There is a specific angle, known as the critical angle, at which an object would fall off a sphere. If the angle is less than the critical angle, the object will stay on the sphere, and if it is greater than the critical angle, the object will fall off.

4. How does the angle at which an object falls off a sphere change with the radius of the sphere?

The angle at which an object would fall off a sphere is directly proportional to the radius of the sphere. This means that as the radius of the sphere increases, the critical angle also increases. For example, if the radius of the sphere is doubled, the critical angle will also double.

5. Can the angle at which an object falls off a sphere be affected by factors other than the radius and gravitational force?

Yes, the angle at which an object would fall off a sphere can be affected by other factors such as air resistance, the surface of the sphere, and the initial velocity of the object. These factors may alter the trajectory of the object and affect the angle at which it falls off the sphere.

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