# Minimum force required to make an object slipping

Let's say we have a when and I exert some force on the center of mass of the wheel. The wheel starts rolling without slipping.At the contact point the static friction force is opposite to the force I am exerting. To find the the minimum force required to make the wheel slip I have to find the torque of the maximum static friction, then find the angular acceleration produced by that torque and then since it will slip the translational acceleration must be greater than that angular acceleration*radius of wheel and solve the equation correct?

## 1 Answer

Yes, it is correct. The maximum static friction is $$F_A^{max}=\mu_S Mg$$, being $$M$$ the mass of the wheel, and its torque is $$\tau_A^{max}=R\mu_S Mg$$, so the maximum angular acceleration without slipping is $$\alpha^{max}=\frac{\tau_A^{max}}{I}=\frac{R\mu_S Mg}{I}$$ Let $$F$$ be the exerted force, which produces an acceleration $$a_{CM}=\frac{F}{M}$$ so $$\frac{F}{M}=a_{CM}=R\alpha^{max}=\frac{R^2\mu_S Mg}{I}\quad\Longrightarrow\quad F^{min}=\frac{R^2\mu_S M^2g}{I}$$