An illustration of the path independence of gravitational work.
Composition Setup
Drop
Ramp
Track
Consider three ways to get a block from height hi down to height hf = 0: dropping the block straight down through the air (freefall), sliding the block down a frictionless ramp, or sliding the block along a frictionless track. The options are illustrated above.
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bigdeck
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Part A
Part A
Assuming that the block starts from rest, what is the final speed of the block as it reaches the ground in each case?
Solution
We begin by solving all three problems using mechanical energy. We will then re-solve the first two cases using the equations of kinematics and dynamics to illustrate the agreement between the methods.
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littledeck
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Method 1
Method 1
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sysa1
System:
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sysa1
Block as plus the earth as a rigid body with infinite mass.
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inta1
Interactions:
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inta1
In each case, there is a conservative gravitational interaction between the block and the earth which will provide a gravitational potential energy. In the final two cases (the ramp and the track) the block is also subject to a non-conservative normal force from the surface upon which it travels. In each case, however, the normal force does no work, since it is always perpendicular to the direction of the block's travel (the block is always moving parallel to the surface and the normal force is, by definition, perpendicular to the surface). Thus, in all three cases, the non-conservative work is zero.
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moda1
Model:
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moda1
.
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appa1
Approach:
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appa1
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inta2
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inta2
External influences from the earth (gravity) and for the block on the ramp, the ramp provides a normal force.
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moda2
Model:
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moda2
One-Dimensional Motion with Constant Acceleration and .
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appa2
Approach:
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appa2
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intb
Interactions:
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intb
External influences from the earth (gravity) and for the block on the ramp, the ramp provides a normal force.
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modb
Model:
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modb
One-Dimensional Kinematics with Constant Acceleration.
