How to solve this problem without using energy considerations? [closed]

Question

This is a problem from my introductory physics textbook:

A cylinder is released from rest from the top of an incline of inclination $\theta$ and length $l$. If the cylinder rolls without slipping, what will be its speed when it reaches the bottom.

This is of course easy to solve using the principle of conservation of mechanical energy. The change in potential energy, $mgl\sin \theta$, must be equal to the kinetic energy at the bottom, i.e., $$\begin{align} mgl\sin \theta &=\frac{1}{2}I\omega ^2 +\frac{1}{2}mv^2 \\ &= \frac{1}{4}mv^2+\frac{1}{2}mv^2 = \frac{3}{4}mv^2 \end{align}$$ Solving for v, $$v=\sqrt{4/3gl\sin \theta}$$

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My question is, how do we solve this problem without using energy considerations, i.e., while taking a force-based approach?

Answer 1

You use force and torque relationships to do the evaluation. You can find this approach discussed in the Halliday and Resnick Physics textbooks.

It is important to recognize that for rolling without slipping $v = r \omega$ and that allows for the relatively simple evaluation using force and torque provided by Ali; this relationship is not true if the object slips.

For a force to do work the force must act through a distance. For rolling without slipping the force of friction does no work because there is no relative motion of the instantaneous point of contact and the surface. Rolling friction provides a force and a torque but does no work. That is why rolling friction has no effect on the energy balance you provided. As another example, for a fluid moving in a pipe assuming the no-slip condition at the pipe walls, the force of friction from the pipe does no work on the fluid.

If the object slips (slides), the energy approach must account for the work done by friction. For a rigid body, all the friction goes into affecting the kinetic energy since there can be no change in the internal energy of a rigid body. You can find the evaluation for the rigid body sliding case- using energy and force/torque- in some physics mechanics textbooks such as Analytical Mechanics by Fowles.

In reality, the object is not rigid and "heating" effects should be considered.

Answer 2

Using torque on the cylinder you can find the angular speed for any time with the given initial conditions but as you can guess it is quite hard to do because you need to solve the differential equation.

Answer 3

The general steps are

1. Make a sketch
2. Draw the forces, torques, coordinate system, etc.
3. Write down Newton's second law for each object (translational/rotational)
4. Derive the equations of motion (position and angle as a function of time)
5. Calculate whatever you need to calculate