An object slides without friction in a bowl shaped like a hemisphere. It is released from rest at one edge of the bowl, slides up the other side before stopping right on the edge of the bowl. During which part of this process is the power being delivered by the gravitational force at is greatest? Not that we want the greatest positive power.
A. At the beginning
B. When the object is at the bottom of the bowl.
C. At the end
D. Sometime between the start and when the object reaches the bottom of the bowl.
E. Sometime between when the object reaches the bottom of the bowl and the end.
I am really not sure about this problem.. My guess is D.
Power delivered by gravity is the product of speed and the gravity (weight) component along the side of the bowl.
Let A be the angle to the object, measured from the center of the hemisphere.
Speed = sqrt(2gRsinA)
Weight component along direction of motion:
M g cos A
Power = M g cosA*sqrt(2 g R sin A)
Power is zero at A = 0 and 90 degrees, and is a maximum somewhere in between. To be positive, the answer must be D.
Thank you for such a wonderful explanation
To determine at which point in the process the power being delivered by the gravitational force is greatest, we need to understand the concept of power and its relationship with work.
Power is defined as the rate at which work is done or energy is transferred. In this case, the power being delivered by the gravitational force is equal to the force of gravity multiplied by the velocity of the object.
At the beginning of the process (option A), the object is released from rest, so its velocity is zero. Since the velocity is zero, the power being delivered by the gravitational force is also zero.
When the object is at the bottom of the bowl (option B), it has reached its maximum speed. Since the velocity is at its maximum, the power being delivered by the gravitational force is also at its maximum. This is because the force of gravity is acting on the object and causing it to accelerate downwards, resulting in a transfer of energy.
At the end of the process (option C), the object has come to a stop on the edge of the bowl. Its velocity is once again zero, so the power being delivered by the gravitational force is zero.
Sometime between the start and when the object reaches the bottom of the bowl (option D), the speed of the object is increasing. As the object gains speed, the power being delivered by the gravitational force also increases.
Sometime between when the object reaches the bottom of the bowl and the end (option E), the speed of the object is decreasing. As the object loses speed, the power being delivered by the gravitational force decreases.
Given these explanations, it can be concluded that the power delivered by the gravitational force is greatest when the object is at the bottom of the bowl (option B). Therefore, your guess is incorrect, and the correct answer is B.