As per the question,
Height ( h ) = 1 m (h)=1m ( h ) = 1 m
Friction coefficient ( μ ) = 0.5 (\mu) = 0.5 ( μ ) = 0.5
Angle of cone ( θ ) = 4 5 ∘ (\theta ) =45^\circ ( θ ) = 4 5 ∘
Let the maximum angular velocity is ω \omega ω
now,
Hence, radius ( r ) (r) ( r ) of the rotating circle,
tan ( 4 5 ∘ ) = r h \tan(45^\circ)=\frac{r}{h} tan ( 4 5 ∘ ) = h r
r = h tan ( 4 5 ∘ ) = 1 r=h\tan (45^\circ) = 1 r = h tan ( 4 5 ∘ ) = 1
So, N = m g cos 4 5 ∘ + m r ω 2 cos 4 5 ∘ N= mg\cos 45^\circ+ mr\omega^2 \cos 45^\circ N = m g cos 4 5 ∘ + m r ω 2 cos 4 5 ∘
= m g 2 + m r ω 2 2 =\frac{mg}{\sqrt{2}}+\frac{mr\omega^2}{\sqrt{2}} = 2 m g + 2 m r ω 2
= m g + m r ω 2 2 =\frac{mg+mr\omega^2}{\sqrt{2}} = 2 m g + m r ω 2
= m ( g + r ω 2 ) 2 =\frac{m(g+r\omega^2)}{\sqrt{2}} = 2 m ( g + r ω 2 )
So, f s = μ N = 0.5 × m ( g + r ω 2 ) 2 = m ( g + r ω 2 ) 2 2 f_s =\mu N = \frac{0.5\times m(g+r\omega^2)}{\sqrt{2}}=\frac{m(g+r\omega^2)}{2\sqrt{2}} f s = μ N = 2 0.5 × m ( g + r ω 2 ) = 2 2 m ( g + r ω 2 )
Hence, m g sin 4 5 ∘ = m ( g + r ω 2 ) 2 2 mg\sin 45^\circ =\frac{m(g+r\omega^2)}{2\sqrt{2}} m g sin 4 5 ∘ = 2 2 m ( g + r ω 2 )
⇒ 2 g = g + r ω 2 \Rightarrow 2g=g+ r\omega^2 ⇒ 2 g = g + r ω 2
⇒ ω 2 = g \Rightarrow \omega^2=g ⇒ ω 2 = g
⇒ ω = 10 = 9.8 r a d / s e c \Rightarrow \omega =\sqrt{10}=\sqrt{9.8} rad/sec ⇒ ω = 10 = 9.8 r a d / sec
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