What is the minimum height you would have to place a solid ball of mass M and radius R (I = 2MR^2 / 5) on a hill in order for the ball to make it completely around a circular loop at the bottom of the hill? The diameter of the loop is D. Assume the ball rolls without slipping.
Height = H
Ke at bottom = m g H =.5 m v^2 + .5 (2mR^2 / 5) (v/R)^2 = total energy
Now to make it around the loop the velocity, u at the top of the loop must be high enough so centripetal acceleration = g
height = D and radius = D/2
(D/2) omega^2 = g
u = R omega
(D/2)u^2/R^2 = g so u^2 = 2 g R^2/D
total energy = m g D + (1/2)mu^2 + (1/2)(2mR^2 / 5)(u^2/R^2)
= m g D + (1/2)m u^2 + (1/5)m u^2
m g H= m g D + (7/10) m u^2
but we know that to just make it u^2 = 2 g R^2/D
h (H-D) = (7/5) (R^2/D)g
check my arithmetic !!!!!
To calculate the minimum height required for the ball to make it completely around the loop, we need to consider the conservation of mechanical energy.
The mechanical energy of the ball at the top of the hill is given by the sum of its potential energy and kinetic energy.
At the bottom of the loop, the ball will have a maximum speed and the normal force acting on it will be zero. This means that the only forces acting on the ball are gravity and the centripetal force.
The potential energy of the ball at the top of the hill is given by mgh, where m is the mass of the ball, g is acceleration due to gravity, and h is the height.
The kinetic energy of the ball at the bottom of the loop is given by (1/2)mv^2, where v is its velocity.
At the top of the hill, the ball will have no velocity, so its kinetic energy is zero.
The minimum height required for the ball to make it completely around the loop can be calculated by equating the potential energy at the top of the hill to the kinetic energy at the bottom of the loop.
Let's go through the calculations step by step:
1. Start with the potential energy at the top of the hill: mgh.
2. At the bottom of the loop, the kinetic energy is given by (1/2)mv^2.
3. Determine the velocity of the ball at the bottom of the loop. Since the ball rolls without slipping, the linear velocity of the ball can be calculated using the formula v = ωr, where ω is the angular velocity and r is the radius of the ball. The angular velocity can be calculated using the formula ω = √(g/R), where g is the acceleration due to gravity and R is the radius of the loop.
4. Substitute the value of ω into the equation for velocity, v = ωr, using the known value of the radius of the loop, D/2.
5. Substitute the value of velocity into the equation for kinetic energy, (1/2)mv^2.
6. Equate the potential energy at the top of the hill to the kinetic energy at the bottom of the loop: mgh = (1/2)mv^2.
7. Simplify the equation by canceling out the mass, and solve for h, the height at the top of the hill.
Following these steps will allow you to calculate the minimum height required for the ball to make it completely around the loop.