Concept explainers
A block of mass 0.500 kg is pushed against a horizon-tal spring of negligible mass until the spring is compressed a distance x (Fig. P8.65). The force constant of the spring is 450 N/m. When it is released, the block travels along a frictionless, horizontal surface to point Ⓐ, the bottom of a vertical circular track of radius R = 1.00 m, and continues to move up the track. The block’s speed at the bottom of the track is vⒶ = 12.0 m/s, and the block experiences an average friction force of 7.00 N while sliding up the track. (a) What is x? (b) If the block were to reach the top of the track, what would be its speed at that point? (c) Does the block actually reach the top of the track, or does it fall off before reaching the top?
(a)
The value of compression in the spring
Answer to Problem 8.65AP
The value of compression in the spring
Explanation of Solution
Given info: The mass of the block is
The formula to calculate the initial kinetic energy of the block is,
Here,
The initial velocity of the block is 0 as it is at rest then the initial kinetic energy of the block is 0.
The formula to calculate the final kinetic energy is,
Here,
The formula to calculate initial potential energy is,
Here,
The formula to calculate the final potential energy is,
Here,
Thus, the final potential energy of the block is
The formula to calculate the initial energy is,
Here,
The final compression distance is 0 as the spring does not move after striking to the block then the final potential energy is 0.
Substitute
Thus, the initial energy is
The formula to calculate the final energy is,
Here,
Substitute
Thus, the final energy is
From the law of conservation of the energy,
Here,
Substitute
Substitute
Rearrange the above formula for
Substitute
Conclusion:
Therefore, the value of
(b)
The speed of the block at the top of the track.
Answer to Problem 8.65AP
The speed of the block at the top of the track is
Explanation of Solution
Given info: The mass of the block is
The formula to calculate the work done by the frictional force is,
Here,
The formula to calculate the initial kinetic energy of the block is,
Here,
The formula to calculate the final kinetic energy is,
Here,
The formula to calculate initial potential energy is,
Here,
The initial height of the block is 0 as the block is at the bottom of the track then the initial potential energy is 0.
The formula to calculate the final potential energy is,
Here,
Substitute
Thus, the final potential energy of the block is
The formula to calculate the initial energy is,
Here,
Substitute
Thus, the initial energy is
The formula to calculate the final energy is,
Here,
Substitute
Thus, the final energy is
The formula to calculate the law of conservation of energy is,
Here,
Substitute
Substitute
Rearrange the above formula for
Substitute
Conclusion:
Therefore, the speed of the block at the top of the track is
(c)
Whether the block reach the top of the track or fall off before reaching the top.
Answer to Problem 8.65AP
The block stays at the top of the track.
Explanation of Solution
Given info: The mass of the block is
The formula to calculate the centripetal acceleration of the block at the top of the track is,
Here,
Substitute
if the centripetal acceleration of the block at the top of the track is less than the acceleration due to gravity then the block fall but centripetal acceleration of the block at the top of the track is greater than the acceleration due to gravity that concludes that the block actually reaches the top of the track.
Conclusion:
Therefore, the block stays at the top of the track.
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Chapter 8 Solutions
EBK PHYSICS FOR SCIENTISTS AND ENGINEER
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