Tuesday, January 22, 2019

Unizor - Physics4Teens - Mechanics - Work and Gravity

Notes to a video lecture on http://www.unizor.com

Work and Gravity

Let's consider a force to lift an object above the planet against the
force of gravity. In simple cases, when the height is relatively small
compared with the size of a planet, the force of gravity is considered
to be constant. In this case we will assume that we lift to substantial
height and have to take into consideration the Law of Gravity that tells
that the force of gravity is proportional to masses of objects involved
(a planet of mass M and an object of mass m that we lift) and inversely proportional to a distance r between the objects:

F = G·M·m /

where G - a gravitational constant.

Our task is to find the work W needed to lift an object of mass m from a surface of a planet of mass M and radius R to height H above its surface.

The easiest approach is to represent all parameters as functions of the distance r between a center of a planet and an object.

Then the force of gravity as a function of r is

F(r) = G·M·m /

An infinitesimal increment (differential) of work equals

dW(r) = F(r)·dr =

= G·(M·m /r²)·

Let's integrate the differential of work on an interval of r [R,R+H]:

[R,R+H]dW(r) =

= G·M·m·
[R,R+H](1/r²)dr =

= G·M·m·
[1/R − 1/(R+H)] =

= G·M·m·H /

For small (relatively to radius of a planet R) height H the expression in curly brackets above is approximately equal to

G·M·m /R² = m·g

where g = G·M / is an acceleration of the free falling on a planet's surface.

By definition, m·g is the weight P of an object on a surface of a planet, which we consider a constant force in this approximation.

So, our formula for work for small height above the planet is reduced to simple expression

W = P·H

which is a base "force times distance" expression that defines the work
in simple case of constant force acting along a trajectory.

For large height H we cannot ignore the change of gravity
as an object moves far from the planet, and the exact formula must be
used to calculate the work.

As in other cases, the work depends only on characteristics of
interacting objects (their masses in this case) and the result of work
(lifting on certain height), not the way how we achieve this result.

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