1-mavzu: kirish. Mexanika bo’limlari. I-mashg’ulotning maqsadi

An Equation for Gravitational Potential Energy

All the vital points about potential energy can be made by focusing onthe example of gravitational potential energy. For simplicity, we treat only vertical motion, and motion close to the surface of the earth, where the
gravitational force is nearly constant. To find an equation for gravitational PE, we examine the case of freefall, in which energy is transformed between kinetic energy and gravitational PE. Whatever energy is lost in one form is gained in an equal amount in the other form, so using the notation KE to stand for KE_f– KE_iand asimilar notation for PE, we have
KE = – PE_grav. (1)
It will be convenient to refer to the object as falling, so that PE is being changed into KE, but the math applies equally well to an object slowing down on its way up. We know an equation for kinetic energy,

KE = 1/2mv², (2)

so if we can relate v to height, y we will be able to relate PE to y, which would tell us what we want to know about potential energy. The y component of the
velocity can be connected to the height via the constant acceleration equation v²= v₀²+ 2a∆y (3)
and Newton’s second law provides the acceleration,
a = F/m, (4) in terms of the gravitational force.
The algebra is simple because both equation (2) and equation (3) havevelocity to the second power. Equation (2) can be solved for v² to give v²=2KE/m, and substituting this into equation (3), we find
2KE_f/m = 2KE_i/m + 2a y.
Making use of equations (1) and (4) gives the simple result
PE_grav= –F y.
[change in gravitational PE resulting from a change in height y; F is the gravitational force on the object, i.e. its
Weight ; valid only near the surface of the earth, where F is constant] ⁶[7]