# Dynamic analysis of cylinder on top of moving plate

Consider the following setup

Everything is initially at rest. $P$ is a constant force which is then applied on the plate for a given time interval. There exists friction between the ground and the plate, and between the plate and the cylinder.

As you have probably guessed, this was part of an exercise, but what I want is to get some intuition on the evolution of the system. Below is my reasoning.

Everything is initially at rest. $P$ is an external force, so it will change the linear momentum of the plate. I know the friction force grows linearly until the maximum static friction force, and then immediately falls to the dynamic friction force. Therefore what determines if the system will move is the maximum static friction force between the ground and the plate, which is

$$F_{fr} = \mu_s \times (m_{plate} + m_{cylinder}) \times g$$

If $P$ is greater than this, we will have movement. Otherwise, the system will remain at rest.

First question: Did I need to consider friction between the ball and the plate for this analysis?

Now that the plate is moving, the cylinder will tend to remain at rest.

Second question: How can I know the motion of the cylinder (I am more concerned with slippage) only from the data that is given?

The friction force that the plate exerts on the cylinder will be directed to the right. Therefore the cylinder will rotate counter-clockwise. Balancing linear impulse and impulse momentum I can write the linear and angular velocity of the cylinder in terms of the friction force. Then, applying the same technique to the plate, I can write the velocity of the plate in terms of the friction force between the cylinder and the plate.

Third question: Why is the friction force between the cylinder and the plate different from $\mu_{k} \times m_{cylinder} \times g$?

Fourth question: To reach the solutions, I equated the assumed the speed of the plate was equal to the speed of the contact point on the cylinder. Why am I allowed to assume this?

The actual values are the following, in case you find them interesting

$m_{plate} = 1.5kg$

$m_{cylinder} = 6kg$

$r_{cylinder} = 0.08m$

$\mu_s = 0.25$

$\mu_k = 0.20$

$P = 25N$

$\Delta t = 0.75s$

• Have you done a free body diagram on the cylinder and, assuming that the coefficient of static friction is not exceeded between the cylinder and the plate, considered both the translational and rotational motion of the cylinder? – Chet Miller Apr 15 '17 at 11:03
• Yes. Using actual values, it yields that the maximum translation speed I can achieve from the friction force over that time interval is lower than the speed attained if I considered the system as only one piece (of mass equal to the sum of the individual masses). Was this what you expected me to see? – user1790813 Apr 15 '17 at 11:56
• Let's see the equations. – Chet Miller Apr 15 '17 at 13:04
• – sammy gerbil Apr 16 '17 at 0:46
• @sammygerbil all those questions assume that the cylinder doesn't slip. In this problem that has to be deduced. Before marking this question as a duplicate, I would like to see at least that answered. – user1790813 Apr 16 '17 at 10:01

2. Adopt a frame of reference which is accelerating with the plate. This requires you to apply a pseudo force $-ma$ acting to the left on the CM of the cylinder. Friction will (partially) oppose this force and create a torque. If the limiting static friction coefficient $\mu_s$ is greater than a critical value, the cylinder will roll; if it is less than the critical value the cylinder will slide and roll. See Accelerated pure rolling and friction for details of how to calculate the critical value of $\mu$.
3. If the cylinder slides, the friction force will be $\mu_k mg$. If the cylinder rolls and accelerates, static friction applies - so friction could be less than the limiting value $\mu_s mg$. Like normal reaction, static friction is a reaction force. It reacts to the pseudo-force by providing sufficient force to create a torque which generates rolling without slipping at the contact point, up to the limit of static friction.
4. I think your assumption is that the contact point remains stationary in the ground frame of reference. This is not a valid assumption. If the cylinder were a block sliding on a frictionless plate, it would accelerate to the left at $-a$ relative to the plate, remaining stationary in the ground frame. However, because rolling objects have greater inertia (compare objects sliding vs rolling down an inclined plane) the acceleration of the CM of the cylinder rolling to the left is less than $a$.