Normal force acting on the billiard ball at point
Answer to Problem 70QAP
Explanation of Solution
Given info:
Billiard ball of,
Radius
Mass
Translational speed of billiard ball at
Point B is at the top of a hill that has a radius of curvature of
Billiard ball rolls without slipping down the track.
Formula used:
Let's name normal force acting on the ball at point
As
Let's name the angular velocity of billiard ball at the top of the track as
Let's name the linear speed of billiard ball at the top of the hill as
Let's name the moment of inertia of billiard ball as
Let's name the vertical distance from potential energy zero level
Conservation of mechanical energy:
Kinetic energy for an object that undergoes both translation and rotation:
Condition for rolling without slipping:
Centrifugal power of billiard ball,
Calculation:
Let's consider the motion of sphere,
Initially the billiard ball is at rest with translational kinetic energy, so
The initial gravitational potential energy is
Final gravitational potential energy is
Conservation of mechanical energy:
Let's consider the kinetic energy
Kinetic energy is part translational and part rotational. We can use
In terms of
Using
Kinetic energy for an object that undergoes both translation and rotation:
Condition for rolling without slipping:
Substitute into kinetic energy equation:
From the general knowledge we know that moment of inertia of a sphere is
So, let's substitute the
Since
Let's consider the billiard ball at the top of the hill,
Centrifugal power of billiard ball,
Substituting
Let's substitute the values,
Conclusion:
Thus, normal force acting on the billiard ball at point
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