# Velocity of a cylinder on a horizontal plane

CSA
Hi, I appreciate any help. Thanks in advance!

## Homework Statement

The image shows the top view of the set-up. Basically, it is a cylinder (resting on a horizontal plane) with two rods on either side attached in line with the centre of mass of the cylinder. Two pieces of string are then taped onto the rods and then tied to another string which is pulled horizontally with a constant force. The string just coils around the rods as the cylinder rolls. Is it possible to determine the velocity of the cylinder after given amount of time? (assuming the force pulling the string is known) Also is it possible to determine what magnitude of force would cause the cylinder to slip, given the coefficient of static friction?

## The Attempt at a Solution

I thought that the horizontal components of the tension in the two strings would create a torque on the two rods which is transferred to the cylinder as it rotates with the rods but, then I ran into more problems like what is the tension in those two strings? and what is the direction of the friction on the cylinder?

#### Attachments

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Mentor
Please show us more details of what you have done so far. For example, what free body diagrams have you drawn.? Have you drawn a free body diagram of the point where the 3 strings meet?

CSA
Hi, thanks for the reply. It is kind of hard to draw how the knot was tied but, its basically a loop at the end of each string and the loop of the main string goes through the other two. Sorry for the messiness. I tried this by thinking of the entire thing as a single object by taking (2T1cosθ as the force as the force, assuming the tensions in both the strings are equal) but I am not sure how to account for the friction on the cylinder and whether this approach actually makes sense.

#### Attachments

Homework Helper
Gold Member
2022 Award
a torque on the two rods
Since the rod radii are unknown, it will be simpler to take it as a force applied at the centre of the cylinder.
(Your diagram shows the force as at the top of the cylinder. How do you get that?)
what is the direction of the friction on the cylinder?
What would happen if there were no friction between cylinder and ground?

CSA
Your diagram shows the force as at the top of the cylinder. How do you get that?
The diagram shows the end of the rod from the side-view. The strings are attached to the top of the rods so they exert a force on them but, the radii of the rods are actually quite small so is it possible to ignore that and take it as though the force is applied at the centre of the cylinder, as you said? (all the dimensions including radius of rods, cylinder, length of strings and mass of cylinder and rods are known). Thanks for helping!

CSA
What would happen if there were no friction between cylinder and ground?

Would the cylinder not roll? Or is it that the strings pulling the rods directly provide a torque to the cylinder?

Homework Helper
Gold Member
2022 Award
The diagram shows the end of the rod from the side-view.
Ok, I had thought it was showing the cylinder.
all the dimensions including radius of rods, cylinder, length of strings and mass of cylinder and rods are known
That is important information, as is the fact that the strings go over the top of the rods.
Would the cylinder not roll? Or is it that the strings pulling the rods directly provide a torque to the cylinder?
The cylinder would rotate, but not necessarily at the precise rate (compared with its linear speed) that would constitute rolling.
So to find out which way the friction acts, you could do that calculation first: if there were no friction, what would the linear and angular accelerations be, and what would that mean for the point of contact with the ground?

• CSA
CSA
Alright I think I can sort it out. Thanks a lot!