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A heavy uniform chain lies on a horizontal table-top. If the coefficient of friction between the chain and table surface is 0.25, then the maximum fraction of length of the chain, that can hang one edge of the table is
- a)20%
- b)25%
- c)35%
- d)15%
Correct answer is option 'A'. Can you explain this answer?
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A heavy uniform chain lies on a horizontal table-top. If the coefficie...
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A heavy uniform chain lies on a horizontal table-top. If the coefficie...
Given, coefficient of friction between chain and table surface, μ = 0.25
Let the length of the chain be L and let x be the maximum fraction of length of chain that can hang one edge of the table.
To find x, we need to find the maximum length of chain that can be kept on the table without hanging down.
Finding the maximum length of chain on the table
Since the chain is uniform, the weight of any part of the chain is proportional to its length.
Let us consider a small part of the chain of length dx at a distance x from the edge of the table.
The weight of this part of the chain is dW = λgdx, where λ is the linear density of the chain, g is the acceleration due to gravity.
The force of friction on this part of the chain is f = μdW = μλgdx
The vertical component of tension in the chain is Tsinθ, where θ is the angle the chain makes with the horizontal.
The horizontal component of tension in the chain is Tcosθ.
At the point where the chain is about to slide, the horizontal component of tension in the chain just balances the force of friction.
Therefore, Tcosθ = μλgdx
The vertical component of tension in the chain just balances the weight of the chain.
Therefore, Tsinθ = λgdx
Dividing these two equations, we get tanθ = 1/μ = 4
Let M be the mass of the chain.
Then, M = λL
The tension in the chain at the edge of the table is T = Mg/2 = λLg/2
Using the equation Tcosθ = μλgdx, we get
Lx/2 * 4/5 = μLdx
Simplifying, we get x = 0.2 = 20%
Therefore, the maximum fraction of length of the chain that can hang one edge of the table is 20%.
Let the length of the chain be L and let x be the maximum fraction of length of chain that can hang one edge of the table.
To find x, we need to find the maximum length of chain that can be kept on the table without hanging down.
Finding the maximum length of chain on the table
Since the chain is uniform, the weight of any part of the chain is proportional to its length.
Let us consider a small part of the chain of length dx at a distance x from the edge of the table.
The weight of this part of the chain is dW = λgdx, where λ is the linear density of the chain, g is the acceleration due to gravity.
The force of friction on this part of the chain is f = μdW = μλgdx
The vertical component of tension in the chain is Tsinθ, where θ is the angle the chain makes with the horizontal.
The horizontal component of tension in the chain is Tcosθ.
At the point where the chain is about to slide, the horizontal component of tension in the chain just balances the force of friction.
Therefore, Tcosθ = μλgdx
The vertical component of tension in the chain just balances the weight of the chain.
Therefore, Tsinθ = λgdx
Dividing these two equations, we get tanθ = 1/μ = 4
Let M be the mass of the chain.
Then, M = λL
The tension in the chain at the edge of the table is T = Mg/2 = λLg/2
Using the equation Tcosθ = μλgdx, we get
Lx/2 * 4/5 = μLdx
Simplifying, we get x = 0.2 = 20%
Therefore, the maximum fraction of length of the chain that can hang one edge of the table is 20%.
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A heavy uniform chain lies on a horizontal table-top. If the coefficie...
Just balance the hanging weight with the friction acting on the remaining part of the rope
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A heavy uniform chain lies on a horizontal table-top. If the coefficient of friction between the chain and table surface is 0.25, then the maximum fraction of length of the chain, that can hang one edge of the table isa)20%b)25%c)35%d)15%Correct answer is option 'A'. Can you explain this answer?
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