A heavy particle is placed at the top of a vertical hoop.Calculate the reaction of the hoop on the particle and the height at which the particle falls off.
To calculate the reaction force of the hoop on the particle, we need to consider the forces acting on the particle at the top of the hoop. At this point, there are two forces acting on the particle:
1. The weight of the particle (mg), which acts vertically downward.
2. The reaction force of the hoop on the particle, which acts perpendicular to the surface of the hoop.
Since the particle is at rest on the hoop, the net force acting on it must be zero. This implies that the reaction force of the hoop on the particle must balance the weight of the particle. Therefore, the magnitude of the reaction force is equal to the weight of the particle (mg).
Now, to find the height at which the particle falls off, we can consider the forces acting on the particle at the highest point of the hoop. When the particle just loses contact with the hoop, the only force acting on it is its weight (mg) pointing vertically downward. At this point, there is no reaction force from the hoop since they are no longer in contact.
Using the concept of conservation of energy, we can equate the initial potential energy of the particle at the top of the hoop (when it starts to fall) to its final kinetic energy when it loses contact with the hoop. Assuming no energy losses due to friction or air resistance:
mgh = 1/2 * mv^2
where m is the mass of the particle, g is the acceleration due to gravity, h is the height from the top of the hoop to where the particle falls off, and v is the velocity of the particle when it loses contact with the hoop.
Simplifying the equation, we can cancel out the mass of the particle:
gh = 1/2 * v^2
From this equation, we can solve for the height (h) at which the particle falls off by knowing the velocity (v) at that point.
To find the velocity, we can use the conservation of mechanical energy:
mgh = 1/2 * mv^2
Canceling out the mass of the particle again:
gh = 1/2 * v^2
Solving for v:
v = √(2gh)
Once we have the velocity, we can substitute it back into the equation for h to find the height at which the particle falls off.