Exercise 2.31 cylinder and hanging mass

A uniform cylinder of mass $M$ sits on a fixed plane inclined at an angle $\theta$ . A string is tied to the cylinder’s rightmost point, and a mass $m$ hangs from the string, as shown in Figure above. Assume that the coefficient of friction between the cylinder and the plane is sufficiently large to prevent slipping.

What is $m$ , in terms of $M$ and $\theta$ , if the setup is static?

#Mechanics #Statics

Easy Math Editor

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Comments

alt text

In the above image, I have marked out the forces on the cylinder, and also, I have removed the mass $\displaystyle m$ .

Firstly, if the system is static, then the tension in the string must be $\displaystyle mg$ , so that there is equilibrium for the mass $\displaystyle m$ .

For attaining rotational equilibrium,
Considering the torque about the contact point of the cylinder and the inclined plane,

$Mg(R\sin\theta) = T(R-R\sin\theta)$

The torque due to the Normal Reaction and the Frictional force, is zero, since those forces pass through the contact point (the point which we are considering the torque about)

$\Rightarrow Mg(R\sin\theta) = mgR(1-\sin\theta)$

$\Rightarrow M\sin\theta = m(1-\sin\theta)$

$\Rightarrow \boxed{m = \frac{M\sin\theta}{1-\sin\theta}}$

Anish Puthuraya - 7 years ago

Anish, which software do you use to draw diagrams?

Karthik Kannan - 7 years ago

I use Photoshop CS5

Anish Puthuraya - 7 years ago

@Anish Puthuraya – Oh, I don't have it. Could you recommend something else with which I can put a decent diagram?

Karthik Kannan - 7 years ago

@Karthik Kannan – Paint is not bad, if you just draw simple diagrams.

Anish Puthuraya - 7 years ago

@Anish Puthuraya – Thanks, Anish! : )

Karthik Kannan - 7 years ago

nice explanation

Hafizh Ahsan Permana - 6 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$