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.
To find the minimum height required for a ball to make it completely around a circular loop, we need to consider the conservation of energy. At the highest point, the ball has gravitational potential energy while at the bottom of the loop, it has gravitational potential energy and kinetic energy.
Let's break down the problem step by step:
Step 1: Calculate the potential energy at the highest point:
The potential energy (PE) at the highest point is given by the equation PE = mgh, where m is the mass of the ball, g is the acceleration due to gravity, and h is the height of the ball from the lowest point of the loop to the highest point.
Step 2: Calculate the potential energy at the bottom of the loop:
At the bottom of the loop, the ball is at its lowest position, so the gravitational potential energy is zero.
Step 3: Calculate the kinetic energy at the bottom of the loop:
The kinetic energy (KE) at the bottom of the loop is given by the equation KE = (1/2)mv^2, where v is the velocity of the ball.
Step 4: Equate the potential energy at the highest point to the sum of potential and kinetic energy at the bottom of the loop:
mgh = (1/2)mv^2
Step 5: Simplify the equation:
Cancel out the mass (m) from both sides of the equation:
gh = (1/2)v^2
Step 6: Use the relationship between velocity and radius at the bottom of the loop:
v = ωr, where ω is the angular velocity and r is the radius of the loop.
Step 7: Express angular velocity in terms of v:
ϖ = v/r, where ϖ is the angular velocity.
Step 8: Simplify the equation by substituting v in terms of ϖ:
gh = (1/2)(ϖr)^2
gh = (1/2)ϖ^2r^2 (please note that r cancels out)
Step 9: Calculate the minimum height (h):
h = (1/2ϖ^2) * r
The minimum height required for the ball to make it completely around the circular loop is h = (1/2ϖ^2) * r, where ϖ is the angular velocity and r is the radius of the loop.