Question

A small block of mass m is placed on the rotating conical surface with semi-vertex angle $\theta$ at the radius r =0.2 m shown in Fig .7-56. Determine the maximum angular velocity $\omega$ of the cone about the vertical axis for which the block will not slip. Friction between block and surface is 0.8

Answer 1

Semi-vertex angle $=\theta$

Radius $r=0.2m$

Angular velocity $=\omega$

Friction between block and surface$=0.8$

Find maximum angular velocity $\omega$ of the come about the vertical axis for which the block will not slip.

Acceleration of a body on an inclined plane.

See image 2

$mg\sin \theta -F^1=mfmf=mg\sin \theta -{\mu }^{1}R=mg(\sin \theta -{\mu }^1\cos \theta )f=g(\sin \theta -{\mu }^1\cos \theta )\frac{V}{t}=g(\sin \theta -{\mu }^1\cos \theta )v=gt(\sin \theta -{\mu }^1\cos \theta )=\omega r=gt(\sin \theta -{\mu }^1\cos \theta )\omega =\frac{gt}{r}(\sin \theta -{\mu }^1\cos \theta )\omega =\frac{gt}{r}(\cos \theta -{\mu }^1\sin \theta )=\frac{10\times t}{0.2}(\cos \theta -0.8\sin \theta )=\frac{10\times t\times 10}{2}(\cos \theta -0.8\sin \theta )=50t(\cos \theta -0.8\sin \theta )$