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A block of mass m is kept on a horizontal ruler. The friction

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A block of mass m is kept on a horizontal ruler. The friction of coefficient between the ruler and the block is $μ$ . The ruler is fixed at one end and the block is at a distance L from the fixed end. The ruler is rotated about the fixed end in the horizontal plane through the fixed end. a. What can the maximum angular speed of the ruler b) if the angular speed of ruler is uniformly increased from zero to an angular acceleration $α$ , at what angular speed will the block slip?

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A block of mass m is kept on a horizontal ruler. The friction of coefficient between the ruler and the block is

μ

. The ruler is fixed at one end and the block is at a distance L from the fixed end. The ruler is rotated about the fixed end in the horizontal plane through the fixed end. a. What can the maximum angular speed of the ruler b) if the angular speed of ruler is uniformly increased from zero to an angular acceleration

α

, at what angular speed will the block slip?

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Text solutionVerified

Sure, here are the step by step solutions: Given, frictional force provides centripetal force i.e.

m L ω^{2} = μ m g

Thus,

ω = \frac{μg}{L}

- Equation (1) Also, net force of circular motion is provided by friction

ω = α t

- Equation (2)

F_{n e t} = m a_{t}^{2} + a_{n}^{2}

μ m g = m L^{2} α^{2} + L^{2} ω^{2}

- Equation (3) Substituting

a_{t} = Lα

and

a_{n} = L ω^{2}

in Equation (3), we get:

μg = L^{2} α^{2} + L^{2} ω^{2}

Substituting

ω = α t

from Equation (2) in the above equation, we get:

μg = L^{2} α^{2} + L^{2} α^{4} t^{4}

Solving for

t

t = \frac{( \frac{μ ^{2} g ^{2} - L ^{2} α ^{2}}{L ^{2} α ^{4}} ) ^{\frac{1}{4}}}{4}

Substituting the value of

t

in Equation (2), we get:

ω = \frac{[ ( \frac{μg}{L} ) ^{2} - α ^{2} ] ^{\frac{1}{2}}}{4}

- Final Answer

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Question Text	A block of mass m is kept on a horizontal ruler. The friction of coefficient between the ruler and the block is $μ$ . The ruler is fixed at one end and the block is at a distance L from the fixed end. The ruler is rotated about the fixed end in the horizontal plane through the fixed end. a. What can the maximum angular speed of the ruler b) if the angular speed of ruler is uniformly increased from zero to an angular acceleration $α$ , at what angular speed will the block slip?
Topic	Circular and Rotation Motion
Subject	Physics
Class	Class 12
Answer Type	Text solution:1

Text solutionVerified

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