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System of Particles and Rotational Motion

Rotational dynamics

A hemispherical bowl of radius R is rotated about its vertical

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Medium

Solving time: 3 mins

A hemispherical bowl of radius $R$ is rotated about its vertical axis. A small particle is kept on its inner surface where the radius makes an angle $θ$ with the vertical. The particle rotates with the bowl without any slipping. The friction coefficient between the block and the bowl surface is $μ$ . Find the range of the angular speed for which the block will not slip.

Views: 5,316 students

Updated on: Oct 12, 2023

Book:

Physics Galaxy Mechanics Vol 1 (Ashish Arora)

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Samuel Discussed

A hemispherical bowl of radius

R

is rotated about its vertical axis. A small particle is kept on its inner surface where the radius makes an angle

θ

with the vertical. The particle rotates with the bowl without any slipping. The friction coefficient between the block and the bowl surface is

μ

. Find the range of the angular speed for which the block will not slip.

9 mins ago

Discuss this question LIVE

9 mins ago

Text solutionVerified

If the bowl rotates at maximum angular speed, the block tends to slip upwards. So, the frictional force acts downward.
Here,

r = R sin θ

From FBD -1

R_{1} - m g cos θ - m ω_{1}^{2} (R sin θ) sin θ = 0

(i) [because

r = R sin θ

]
and

μ R_{1} m g sin θ - m_{1}^{2} (R sin θ) cos θ = 0

(ii)
Substituting the value of

R_{1}

from

Eq

(i) in Eq(ii). it can be found out that

ω_{1} = [\frac{g ( s i n θ + μ c o s θ )}{R s i n θ ( c o s θ - μ s i n θ )}]^{1/2}

Again, for minimum speed, the frictional force

μ R_{2}

acts upward. From FBD-2, it can be proved that,

ω_{2} = [\frac{g ( s i n θ - μ c o s θ )}{R s i n θ ( c o s θ + μ s i n θ )}]^{1/2}

∴

the range of speed is between

ω_{1}

and

ω_{2}

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Question Text	A hemispherical bowl of radius $R$ is rotated about its vertical axis. A small particle is kept on its inner surface where the radius makes an angle $θ$ with the vertical. The particle rotates with the bowl without any slipping. The friction coefficient between the block and the bowl surface is $μ$ . Find the range of the angular speed for which the block will not slip.
Updated On	Oct 12, 2023
Topic	System of Particles and Rotational Motion
Subject	Physics
Class	Class 11
Answer Type	Text solution:1 Video solution: 5
Upvotes	593
Avg. Video Duration	10 min

Text solutionVerified

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