Question

A coin is pushed down tangentially from an angular position $\theta$ on a cylindrical surface, with a velocity v as shown in the Fig. If the coefficient of friction between the coin and surface is $\mu$, find friction between the tangential acceleration of the coin.

Answer 1

As the coin slides down, friction is kinetic and acts up along the plane.

Equation of motion:

$\sum F_r=N-mgcos\theta =ma$, (i)

$\sum F_t=mgsin\theta -f_k=m{a}_t$ (ii)

But $f_k=\mu N$ (ii)

And centripetal accleration is $a_r=\frac{v^2}{R}$ (iv)

Substituting $f_k$ from Eq. (iii) and $a_r$ from Eq. (iv), in Eq. (ii),

We get

$a_t=gsin\theta -\frac{\mu N}{m}$ (v)

Now, substituting N from Eq. (i) in Eq. (v), we get

Now, substuting N from Eq. (i) in Eq. (v), we get

$a_t=g(sin\theta -\mu cos\theta )-\frac{\mu v^2}{R}$