A point mass m starts from rest and slides down the surface of a frictionless solid sphere of radius r. Find the angle at which the mass flies off the sphere
To find the angle at which the mass flies off the sphere, we need to consider the forces acting on the mass as it slides down.
When the mass is at the topmost point of the sphere, the only force acting on it is the weight, which points downward. As the mass slides down, the normal force from the sphere surface provides the necessary centripetal force to keep the mass moving in a circular path.
At the point where the mass flies off, the normal force becomes zero, and therefore, the centripetal force is also zero. At this point, only the weight acts on the mass.
We can analyze the forces acting on the mass at the topmost point of the sphere to determine the conditions for the mass to fly off:
1. Weight (mg): This force points downward and can be decomposed into two components: one parallel to the sphere surface and one perpendicular to it.
2. Normal force (N): This force acts perpendicular to the sphere surface, providing the necessary centripetal force.
3. Centripetal force (Fc): This force acts inward, keeping the mass moving in a circular path.
At the topmost point, the normal force and the weight are in the same direction (opposite to the centripetal force).
To find the angle at which the mass flies off, we need to equate the centripetal force and the weight component parallel to the sphere surface.
1. Weight component parallel to the surface:
mg * sinθ = Fc
2. Centripetal force:
Fc = m * (v^2 / r)
Combining these equations, we get:
mg * sinθ = m * (v^2 / r)
The mass cancels out, and we are left with:
g * sinθ = v^2 / r
To solve for the angle θ, we need to find the velocity v at the point where the mass flies off. We can use conservation of mechanical energy to determine this.
At the topmost point, all the initial potential energy (mgh) is converted into kinetic energy (1/2mv^2).
1/2mv^2 = mgh
v^2 = 2gh
Substituting this value of v^2 into the earlier equation:
g * sinθ = 2gh / r
Simplifying, we find:
sinθ = 2h / r
Finally, we can find the angle θ by taking the inverse sine (sin^(-1)) of both sides:
θ = sin^(-1)(2h / r)
This equation gives the angle at which the mass will fly off the sphere.