Concept explainers
(a)
The speed at the bottom of the half-pipe.
(a)
Answer to Problem 11.48AP
The speed at the bottom of the half-pipe is
Explanation of Solution
Given info: The mass of particle is
Write the expression for conservation of work and energy law.
Here,
The skateboarder is at rest at point A, so there is a potential energy at point A,
Here,
The center of mass moves through one quarter of the circle.
The radius of the circle is,
The skateboarder is in motion so it acquires the kinetic energy at point B,
Here,
Substitute
Substitute
Conclusion:
Therefore, the speed at the bottom of the half-pipe is
(b)
The
(b)
Answer to Problem 11.48AP
The angular momentum of him about the center of curvature at the point B is
Explanation of Solution
Given info: The mass of particle is
Write the expression for the angular momentum about the center of curvature.
Here,
Substitute
Conclusion:
Therefore, the angular momentum of him about the center of curvature at the point B is
(c)
To explain: The angular momentum of him is constant in this maneuver, whereas the kinetic energy of his body is not constant.
(c)
Answer to Problem 11.48AP
After the passing point B, there is no torque about the axis of the channel act on him so; the angular momentum will be constant, but his legs convert the chemical energy into mechanical energy and the kinetic energy of his body is not constant.
Explanation of Solution
Given info: The mass of particle is
A skateboarder passes the point B, so there is no tangential force acts on him because the wheels on the skate prevent this force. The torque is zero due to no tangential force, so the angular momentum will be constant.
The kinetic energy increase because his legs convert chemical energy into mechanical energy and the kinetic energy will not be constant. While the normal force rises trajectory to enhance his linear momentum.
Conclusion:
Therefore, after the passing point B, there is no torque about the axis of the channel act on him so; the angular momentum will be constant, but his legs convert the chemical energy into mechanical energy and the kinetic energy of his body is not constant.
(d)
The speed immediately after the skateboarder stands up.
(d)
Answer to Problem 11.48AP
The speed of skateboarder after he stands up is
Explanation of Solution
Given info: The mass of particle is
The skateboarder stands up, so the distance is,
Write the expression for angular momentum.
Here,
Substitute
Conclusion:
Therefore, the speed of skateboarder after he stands up is
(e)
The amount of chemical energy in the skateboarder’s leg was converted into mechanical energy in skateboarder-Earth system when he stood up.
(e)
Answer to Problem 11.48AP
The amount of chemical energy in the skateboarder’s leg was converted into mechanical energy in skateboarder-Earth system when he stood up is
Explanation of Solution
Given info: The mass of particle is
At point B, the skate boarder has kinetic and chemical energy is,
Here,
At point C, he has kinetic energy due and the potential energy is,
Here,
Write the expression of the conservation of energy.
Substitute
Write the expression for the kinetic energy at point B.
Substitute
Thus, the kinetic energy at point B is
Write the expression for the kinetic energy at point C.
Substitute
Thus, the kinetic energy at point C is
Write the expression for potential energy at point C.
Here,
The radius of the pipe at point C,
Substitute
Thus, the potential energy at point C is
Substitute
Conclusion:
Therefore, the amount of chemical energy in the skateboarder’s leg was converted into mechanical energy in skateboarder-Earth system when he stood up is
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