A block of ice of mass m rests on the very top of a smooth metal hemishpere. The tiniest nudge sends it sliding down the side of the hemisphere. The acceleration of gravity is g and the radius of the hemisphere is R. At what angle, with respect to the vertical, does the block fly off the surface of the hemisphere?
To determine the angle at which the block of ice flies off the surface of the hemisphere, we can first analyze the forces acting on the block.
1. Gravitational force (mg): This force acts vertically downwards and is equal to the product of the mass of the block (m) and the acceleration due to gravity (g).
2. Normal force (N): This force acts perpendicular to the surface of the hemisphere and counteracts the gravitational force. Since the block is at rest initially, the normal force must be equal in magnitude and opposite in direction to the gravitational force (mg).
3. Frictional force (f): This force opposes the motion of the block. As the block slides down the surface of the hemisphere, the frictional force points upwards along the tangent to the surface of the hemisphere.
Given that the block is on a smooth metal hemisphere, there is no friction acting initially. However, as soon as the block reaches the critical angle where it is about to leave the surface, the friction force becomes significant and influences its motion.
At the critical angle, the normal force becomes zero, allowing the block to fly off. Therefore, we need to find the angle at which the normal force becomes zero.
Using trigonometry, we can determine the relationship between the angle and the forces acting on the block. Consider a free-body diagram of the forces acting on the block when it is about to leave the surface:
/|
/ |
N /__| mg
| |
| |
f |
<----|
In the perpendicular direction, we have:
N - mg * cosθ = 0,
N = mg * cosθ.
In the tangential direction, we have:
mg * sinθ - f = 0,
f = mg * sinθ.
When the normal force N becomes zero, we have:
mg * cosθ = 0,
cosθ = 0,
θ = 90°.
Therefore, the block flies off the surface of the hemisphere at a right angle, perpendicular to the vertical.
Note: Since the problem states that the block is nudged, we assume that it receives enough initial velocity or impulse to overcome the static friction and start moving.