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Chapter 10: Problem 61

Starting from rest, a hollow ball rolls down a ramp inclined at angle \(\theta\) to the horizontal. Find an expression for its speed after it's gone a distance \(d\) along the incline.

Short Answer

Expert verified
The speed \(v\) of the ball after travelling a distance \(d\) along the incline is \(v = \sqrt{(6/5)gdsin(\theta)}\).

Step by step solution

01

Understanding the problem

In this problem, the hollow ball starts rolling down from the rest due to gravitational potential energy. As it rolls down, potential energy gets converted into kinetic energy due to its motion.
02

Determine the Potential Energy initially

Potential energy at initial height is given by \(PE = mgh\), where \( m\) is the mass of the ball, \(g\) is the acceleration due to gravity, \( h\) is the initial height. The height \(h\) can be determined from the inclined distance \(d\) and the angle of the incline \(\theta\), which is \(h=dsin(\theta)\). So \(PE=mgdsin(\theta)\).
03

Determine the Kinetic Energy

For a hollow sphere, when it is rolling without slipping, the kinetic energy (KE) is given by \(KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2\), where \(v\) is the velocity of the sphere, \(I\) is the moment of inertia and \(\omega\) is the angular velocity. For a hollow sphere, \(I = \frac{2}{3}mr^2\), where \(r\) is the radius and \(\omega = \frac{v}{r}\). Substituting these values,we get \(KE = \frac{k}{m}v^2\), where \(k=\frac{5}{3}\).
04

Applying Conservation of Energy

According to the law of conservation of energy, the kinetic energy at the bottom of the ramp must equal the potential energy at the top, i.e. \(PE = KE\). Substituting the expressions obtained from Step 2 and Step 3, we get \(mgdsin(\theta) = \frac{5}{3}mv^2\).
05

Solve for v

Finally, solve the equation obtained in Step 4 for \(v\), the speed of the ball. The mass \(m\) cancels out. Solving gives \(v = \sqrt{(6/5)gdsin(\theta)}\).

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