Concept explainers
(a)
The speed of the sled and rider at point C.
(a)
Answer to Problem 77AP
The speed of slader and rider at point C is
Explanation of Solution
Consider sled-chute-earth as an isolated system. Since, there is no friction force acting on the system.
Write the eqution for conservation of energy from point A to point C.
Write the equation for conservation of energy
Here,
Since, the system has potential energy due to gravitation of earth.
Write the equation for gravitational potential energy.
Here,
Write the expression for change in potential energy of the system.
Here,
Substitute
Simplify the above equation.
Here,
Since, the system has kinetic energy due to motion of sled and rider.
Write the equation for kinetic energy.
Here,
Write the expression for the change in kinetic energy.
Here,
Substitute
Simplify the above equation.
Here,
Substitute
Rearrange the above equation for
Conclusion:
Substitute
Thus, the speed of slader and rider at point C is
(b)
The magnitude of the total force the water exerts on the sled.
(b)
Answer to Problem 77AP
The magnitude of the total force the water exerts on the sled is
Explanation of Solution
Consider sled-water as a system.
Since, the friction force exerted by water is in retarding force and hence non-conservative force.
Write the equation for conservation of energy
Here,
Write the equation for work done by retarding force.
Here,
Substitute
Rearrange the above equation for
Simplify the above equation.
Since, the total force exerted by the water is friction force and normal force acting on the sled.
Write the equation for normal force
Here,
Write the expression for the magnitude of the total force
Here,
Since, the normal force and friction force are exerted in perpendicular direction.
Substitute
Simplify the above equation.
Conclusion:
Subsitute
Subsitute
Substitute
Thus, the magnitude of the total force the water exerts on the sled is
(c)
The magnitude of the force the chute exerts on the sled at point B.
(c)
Answer to Problem 77AP
The magnitude of the force the chute exerts on the sled at point B is
Explanation of Solution
Consider the sled on the chute at the point B as shown in figure (a).
Write the expression for the angle
Here,
Since, there is no motion in perpendicular direction fo the motion of sled, hence the net force at the point B will be zero.
Write the expression for net force in y-direction as shown in figure (I).
Here,
Conside the free body diagram of the sled and rider at point B.
Write the expression for net force.
Substitute
Rearrange the above equation for
Conclusion:
Substitute
Substitute
Thus, the magnitude of the force the chute exerts on the sled at point B is
(d)
The force exerted by the chute on the sled at point C where the chute is curving in the vertical plane.
(d)
Answer to Problem 77AP
The force exerted by the chute on the sled at point C where the chute is curving in the vertical plane is
Explanation of Solution
Consider the chute is curving in the vertical plane at point C.
The free body diagram of the sled at point C is as shown in figure (b).
Since, the sled has normal force and the centripital force is outward the center of curve.
Write the equation for net force at point C.
Here,
Substitute
Rearrange the above equation.
Conclusion:
Subsitute
Thus, the force exerted by the chute on the sled at point C where the chute is curving in the vertical plane is
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