Concept explainers
(a)
The speed of the block when it reaches the bottom of the curve.
(a)
Answer to Problem 77PQ
The speed of the block when it reaches the bottom of the curve is
Explanation of Solution
The energy conservation equation for a system is,
Here,
There is no friction during the motion so that the change in thermal energy will also be zero (
The total potential energy is the sum of the gravitational potential energy and the elastic potential energy.
Using the expression for the total potential energy of the system in the initial and final condition to equation (II) yields,
The initial kinetic energy of the block is zero and at the bottom of the track where
Write the expression for the final kinetic energy of the block.
Here,
Write the expression for the initial gravitational potential energy of the block.
Here,
use equation (VI) and (VII) in equation (V) and solve for
Conclusion:
Substitute
Therefore, the speed of the block when it reaches the bottom of the curve is
(b)
The magnitude of the
(b)
Answer to Problem 77PQ
The magnitude of the friction force acting on the block is
Explanation of Solution
Applying the conservation of energy condition from the moment the block reaches the bottom of the track until it finally stops. While there are no external forces performing work on the block-track system during this motion (
The initial kinetic energy in the horizontal path in the bottom is equal to the final kinetic energy calculated in part (a) that corresponds to a speed of
Modify equation (IX) using the expression for the kinetic energy.
Write the expression for the change in thermal energy.
Here,
Use equation (X) in equation (XI), and solve for
Conclusion:
Substitute
Therefore, the magnitude of the friction force acting on the block is
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