
Concept explainers
(a)
The distance traveled by block along the incline plane.
(a)

Explanation of Solution
Given:
The mass of block is
The initial speed of block is
The coefficient of friction is
The angle of incline with the horizontal is
Formula used:
Write the expression for friction force along the incline.
Here,
Write the expression for thermal energy.
Here,
Substitute
Write the expression for change in potential energy.
Here,
Write the expression for change in kinetic energy.
Here,
Total energy of block is conserved at all points. Work done by external force is equal to the sum of change in gravitational potential energy, kinetic energy and thermal energy.
Write the expression of work done by external force.
Here,
There is no external force acting on the block; so, the work done by external force is zero.
Substitute
Substitute
Calculation:
Substitute
Conclusion:
Thus, the distance traveled by block along the incline is
(b)
The speed of the block when it has traveled half the distance in part (a).
(b)

Explanation of Solution
Given:
The mass of block is
The initial speed of block is
The coefficient of friction is
The angle of incline with the horizontal is
Formula used:
Write the expression for friction force along the incline.
Here,
Write the expression for thermal energy.
Here,
Substitute
Write the expression for change in potential energy.
Here,
Write the expression for change in kinetic energy.
Here,
Total energy of block is conserved at all points. Work done by external force is equal to the sum of change in gravitational potential energy, kinetic energy and thermal energy.
Write the expression of work done by external force.
Here,
There is no external force acting on the block; so, the work done by external force is zero.
Substitute
Substitute
For object velocity at halfway on incline plane:
Substitute
Substitute
Calculation:
Substitute
Conclusion:
Thus, the speed of the block when it has traveled half the distance in part (a)is
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