Question

Figure shows a vertical force $F$ that is applied tangentially to a uniform cylinder of weight $W$. The coefficient of static friction between the cylinder and all surfaces is $0.5$. Find the maximum force in terms of $W$ that can be applied without causing the cylinder to rotate.

Answer 1

Since the cylinder is not rolling,

Let axis of rotation be at $A$

Torque, $\tau =I\alpha$

Since $\alpha =0\tau =0F\times R+N_1\times 0+\mu N_1\times 0=\mu N_2\times R+N_2\times RF=(1+\mu )N_2⟶\left(1\right)$

Balancing vertical forces

$\mu N_2+F+N_1=mg⟶\left(2\right)$

Balancing horizontal forces

$\mu N_1=N_2⟶\left(3\right)$

From $3$

$N_1=\frac{N_2}{\mu }\mu N_2+F+\frac{N_2}{\mu }=mg{N}_2=\frac{\mu mg-F\mu }{{\mu }^2+1}⟶\left(4\right)$

From $4$

$F=\frac{(1+\mu )(\mu mg-F\mu )}{{\mu }^2+1}F=\frac{(1+\mu )(1+\mu )}{2{\mu }^2+\mu +1}F=\frac{\mu W.(1+\mu )}{2{\mu }^2+\mu +1}$