Energy Comparison in a Frictionless Track System
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A block is held at rest against a compressed spring at point $A$ at the top of a frictionless track of height $h$, as shown above. The block is released, loses contact with the spring at point $B$, and slides along the track until it passes point $C$, also at height $h$. How do the potential energy $U$ of the block-Earth system and the kinetic energy $K$ of the block at point $C$ compare with those at point $A$?
Potential Energy of Block-Earth System | Kinetic Energy of Block
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(A) $U_C = U_A$ | $K_C = K_A$
(B) $U_C = U_A$ | $K_C > K_A$
(C) $U_C > U_A$ | $K_C = K_A$
(D) $U_C > U_A$ | $K_C > K_A$
This question includes visual content: A diagram shows a frictionless track with two elevated flat sections at height $h$ connected by a dip. On the left section, a block is at point $A$ pressed against a horizontal compressed spring. Point $B$ is slightly to the right of $A$ on the same elevated section. The track then curves down and back up to another flat section on the right, where point $C$ is located at the same height $h$. Vertical arrows labeled $h$ indicate the height of both the left and right flat sections from a dashed baseline.
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Step by Step Written Solution
In this problem, we have a block held against a compressed spring at point A on a frictionless track. We need to compare the gravitational potential energy and kinetic energy at point C with their values at point A.
Conservation of Energy Analysis
First, let's look at the gravitational potential energy of the block-Earth system. The formula for potential energy is mass times gravity times height.
The problem states that point A is at height h. Therefore, the potential energy at point A is m g h.
Now look at point C. It is also at height h. Since the height hasn't changed relative to the reference level, the potential energy must be the same.
So our first conclusion is that U sub C equals U sub A. This narrows our options down to A or B.
Next, let's consider the kinetic energy. At point A, the block is held at rest, so its initial kinetic energy is zero.
Kinetic Energy Comparison
At point A, the system also has elastic potential energy from the compressed spring. Let's call this U sub s.
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