
Concept explainers
The magnitude of the acceleration of the block resting on an inclined plane.

Answer to Problem 80QAP
The magnitude of the acceleration of the block resting on an inclined plane is 3.77 m/s2.
Explanation of Solution
Given info:
Mass of the block placed on the incline
Mass of the hanging block
Angle made by the plane with horizontal
Coefficient of static friction
Coefficient of kinetic friction
Formula used:
Free body diagrams are drawn for the two blocks and the acceleration of the block is determined using the force equations for both the blocks.
The free body diagram for the block of mass
Since m2 is greater than m1, the hanging block would tend to move down and the block on the incline would slide upwards along the incline.
The weight of the block is
The total force acting along the +x direction is given by,
Here, ax is the block s acceleration along the downward direction (+x ).
Draw the free body diagram for the block of mass
The weight
Resolve the weight
Both the blocks have the same magnitude of acceleration.
The force equation along the +x direction is given by,
The force equation along the +y direction is given by,
Since the block is in equilibrium in the y direction,
Hence,
The force of friction and the normal force are related as follows:
The value of the coefficient of friction
Calculation:
First determine, if the system is at rest or in motion.
The system will be at rest if
If the system is at rest, equation (1) can be written as,
Since
Substitute the values of the variables in the above equation,
Calculate the value of
The component
If the system is at rest, assume the maximum force of static friction to act on the block.
Then,
From equation (4),
The component
Calculate the maximum force of static friction acting on the block.
Therefore,
Since it is seen that
Add equations (1)
Substitute the expressions for w2,w1x and fk in the expression and write an expression for ax.
Substitute the values of the variables in the expression and determine the magnitude of the acceleration of the block on the incline.
Conclusion:
The magnitude of the acceleration of the block resting on an inclined plane is 3.77 m/s2.
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