A small metal washer is placed on the top of a hemisphere of radius R. What minimum horizontal velocity should be imparted to the washer to detach it from the hemisphere at the initial point of motion?

To determine the minimum horizontal velocity required for a small metal washer to detach from the top of a hemisphere of radius R, we need to analyze the forces acting on the washer as it moves along the surface of the hemisphere. The key here is to understand the concept of centripetal force and the conditions for the washer to lose contact with the hemisphere.

Understanding Forces at Play

When the washer is placed on top of the hemisphere, it is subject to gravitational force acting downwards and the normal force from the surface of the hemisphere acting perpendicular to the surface. As the washer begins to move horizontally, it will also experience a centripetal force due to its circular motion along the hemisphere's surface.

Key Concepts

• Gravitational Force (Weight): This is the force acting downwards on the washer, given by mg, where m is the mass of the washer and g is the acceleration due to gravity.
• Normal Force: This force acts perpendicular to the surface of the hemisphere and will vary as the washer moves.
• Centripetal Force: Required to keep the washer moving in a circular path, given by mv²/R, where v is the velocity of the washer and R is the radius of the hemisphere.

Condition for Detachment

The washer will detach from the hemisphere when the normal force becomes zero. At this point, all the gravitational force is used to provide the necessary centripetal force. Therefore, we can set up the following equation at the point of detachment:

When the washer is at an angle θ from the vertical, the forces can be analyzed as follows:

• The component of gravitational force acting towards the center of the hemisphere is mg \cos(θ).
• The required centripetal force is mv²/R.

Setting these equal gives us:

mg \cos(θ) = mv²/R

Finding the Angle of Detachment

For the washer to just lose contact, we consider the point where it is at the edge of the hemisphere, which occurs when θ is such that the normal force is zero. This typically happens at an angle of θ = 90° (the washer is about to fall off). At this point, the equation simplifies to:

mg = mv²/R

Calculating Minimum Velocity

From the equation above, we can cancel m (assuming the mass is not zero) and rearrange to find the minimum velocity:

g = v²/R

Thus, solving for v gives:

v² = gR

Taking the square root of both sides, we find:

v = √(gR)

Conclusion

The minimum horizontal velocity that should be imparted to the washer to detach it from the hemisphere at the initial point of motion is given by the formula v = √(gR). This means that if you want the washer to just lose contact with the hemisphere, you need to give it a horizontal velocity equal to the square root of the product of the acceleration due to gravity and the radius of the hemisphere.