To determine the conditions under which the hoop will roll without bouncing when the small body A reaches the lowest point, we need to analyze the forces and motions involved. The key concept here is the balance between gravitational force, centripetal force, and the hoop's motion. Let's break this down step by step.
Understanding the System
We have a thin rigid hoop of radius R and mass equal to that of the small body A, which is fixed inside the hoop. When the hoop rolls on a horizontal surface, it experiences various forces, particularly when the body A is at the lowest point of the hoop.
Forces Acting on the Hoop
At the lowest position of the hoop, the following forces come into play:
- Gravitational Force (Weight): This acts downward on body A and the hoop.
- Normal Force: This is the force exerted by the ground on the hoop, acting upward.
- Centripetal Force: As the hoop rolls, body A experiences a centripetal force due to its circular motion.
Condition for No Bouncing
For the hoop to roll without bouncing, the normal force must be sufficient to counteract the gravitational force acting on body A. When the hoop is rolling, the centripetal force required to keep body A in circular motion must also be considered. The condition can be expressed mathematically.
Mathematical Analysis
At the lowest point, the centripetal force (Fc) required for body A can be expressed as:
Fc = (m * v^2) / R
Where:
- m is the mass of body A (equal to the mass of the hoop).
- v is the velocity of the center of mass of the hoop.
- R is the radius of the hoop.
The gravitational force (Fg) acting on body A is:
Fg = m * g
Where g is the acceleration due to gravity.
Setting Up the Equation
For the hoop to roll without bouncing, the normal force (N) must equal the gravitational force minus the centripetal force:
N = Fg - Fc
Substituting the expressions for Fg and Fc, we get:
N = mg - (m * v^2) / R
For the hoop to not bounce, the normal force must be greater than or equal to zero:
mg - (m * v^2) / R ≥ 0
Simplifying the Condition
Dividing through by m (assuming m ≠ 0), we find:
g ≥ (v^2) / R
Rearranging gives us:
v^2 ≤ gR
Taking the square root of both sides, we find the maximum velocity:
v ≤ √(gR)
Conclusion
Thus, for the hoop to roll without bouncing when body A is at the lowest point, the velocity of the center of the hoop, v0, must satisfy the condition:
v0 ≤ √(gR)
This means that as long as the velocity of the hoop is less than or equal to the square root of the product of gravitational acceleration and the radius of the hoop, it will roll smoothly without bouncing. If the velocity exceeds this value, the hoop will experience bouncing due to insufficient normal force to counteract the gravitational pull on body A.
