By how much does the KE of a mass of 6 kg change as the elevation of the mass changes by -5.3 meters, assuming that the only force acting on the mass is gravitational? What is the associated PE change?
The PE change is the work that must be done against gravity as the object changes elevation. The force exerted on the object by gravity is its weight. If we regard the positive direction as upward, then the weight is:
This force does work on the object. The work done BY the object against gravity is the negative of the work gravity does ON the object.
- work by gravity = - 58.8 Newtons * (-5.3 meters) = --311.64 Joules,
so that the work done by the system against gravity is
- work by system against gravity = -(--311.64 Joules) = -311.64 Joules.
We can understand this easily enough. If you lift something you are doing work against gravity. Whether you are doing the lifting or not, when an object is raised work is done against gravity. When this happens the gravitational PE of the object increases by an amount equal to the work required to lift it.
Thus the work done by the system against gravity is equal to the change in its gravitational PE. In this case the work is negative; the PE decreases as the object moves lower.
If no other forces act on the system, then this gravitational PE must come at the expense of the system’s KE, so the KE of the system changes by
This is also not difficult to understand. An object case rise because someone is lifting it, or it can rise because it's already moving upward with a sufficient velocity, or it can divert some of its KE into upward motion (e.g., a glider). In either of the latter two cases the object rises by converting KE to PE.
The work required to change the elevation of an object by `dy is equal to the product of the force exerted against gravity and `dy.
weight = m * g, the force exerted on the mass of the object by gravity.
work against gravity = m g `dy.
In this problem the change in elevation is negative, so the work against gravity and hence the change in PE is negative.
If no forces other than gravity act on the object, then the work done against gravity is equal and opposite to its change in KE.
An increase in PE would come at the expense of KE (e.g. a rising object under the influence of only gravity slows as its PE increases); a decrease in PE (corresponding to a falling object) increases the KE (falling objects under the influence of only gravity speed up).
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