Answer to Question #127843 in Classical Mechanics for Shreyansh

Question #127843
A particle inside the rough surface of a rotating cone about its axis is at rest, relative to it as a height of 1m above its vertex. Friction coefficient is μ=0.5. If half angle of cone is 45°, the maximum angular velocity of revolution of cone can be :
1
Expert's answer
2020-08-02T15:10:46-0400

As per the question,

Height (h)=1m(h)=1m

Friction coefficient (μ)=0.5(\mu) = 0.5

Angle of cone (θ)=45(\theta ) =45^\circ

Let the maximum angular velocity is ω\omega

now,


Hence, radius (r)(r) of the rotating circle,

tan(45)=rh\tan(45^\circ)=\frac{r}{h}

r=htan(45)=1r=h\tan (45^\circ) = 1

So, N=mgcos45+mrω2cos45N= mg\cos 45^\circ+ mr\omega^2 \cos 45^\circ

=mg2+mrω22=\frac{mg}{\sqrt{2}}+\frac{mr\omega^2}{\sqrt{2}}

=mg+mrω22=\frac{mg+mr\omega^2}{\sqrt{2}}

=m(g+rω2)2=\frac{m(g+r\omega^2)}{\sqrt{2}}

So, fs=μN=0.5×m(g+rω2)2=m(g+rω2)22f_s =\mu N = \frac{0.5\times m(g+r\omega^2)}{\sqrt{2}}=\frac{m(g+r\omega^2)}{2\sqrt{2}}

Hence, mgsin45=m(g+rω2)22mg\sin 45^\circ =\frac{m(g+r\omega^2)}{2\sqrt{2}}

2g=g+rω2\Rightarrow 2g=g+ r\omega^2

ω2=g\Rightarrow \omega^2=g

ω=10=9.8rad/sec\Rightarrow \omega =\sqrt{10}=\sqrt{9.8} rad/sec


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