Review Chap. 7 Potential Energy and Energy Conservation PHYS 218 sec. 517-520 Review Chap. 7 Potential Energy and Energy Conservation

Gravitational potential energy Energy associated with the position of bodies in a system A measure of the potential or possibility for work to be done The work done on the object to change its position is stored in the object in the form of an energy Gravitational potential energy y2 y1 When the body moves up, the work done by the gravitational force is negative and the potential energy increases.

Potential energy The potential energy is a relative quantity. You have to specify the reference point when you define the potential energy. For example, the gravitational potential energy is mgy and the point where y = 0 should be specified, which is the reference point when you define the potential energy. What is physically meaningful is the change of the potential energy. The absolute value does not have physical meaning. Note that the work done by a force is equal to the negative of the potential energy change.

Conservation of mechanical energy So, K + U is conserved This defines the total mechanical energy of the system. Conservation of mechanical energy

Height of a baseball from energy conservation Ex 7.1 Height of a baseball from energy conservation Energy conservation is very useful to obtain speed or position, in particular, when it its very difficult to use Newton’s laws of motion.

When forces other than gravity do work Therefore, for example, if there is friction force, the total mechanical energy is not conserved. Instead, we have

Work and energy in throwing a baseball Ex 7.2 Work and energy in throwing a baseball From y = 0 to y1 State 3 State 2 From y = y1 to y2 State 1 Two solutions; moving up and moving down Here we use different choice for the y-axis from the textbook. But the final answers are the same as it should be.

Gravitational potential energy for motion along a curved path Only Dy contributes The total work becomes The total work done by the gravitational force depends only on the difference in height

Calculating speed along a vertical circle Ex 7.4 Calculating speed along a vertical circle Point 1 Speed at the bottom of the ramp What we need is speed not velocity, so v2 is positive Point 2 Normal force at the bottom of the curve

Vertical circle with friction: same as Ex 7 Vertical circle with friction: same as Ex 7.4 but there is friction force. What is the work done by the friction force? Ex 7.5

Inclined plane with friction Ex 7.6 Inclined plane with friction Motion of a crate: Point 1 (speed v1) g Point 2 (speed v2 = 0) g Point 3 (speed v3) Point 2 equal to Point 1 Magnitude of a constant friction force Point 1, 3 Consider the motion from Point 1 to Point 2

Consider the motion from Point 1 to Point 3 Speed at Point 3 Consider the motion from Point 1 to Point 3 Be careful: Speed is always positive while velocity can be negative What we want is speed not velocity, so v3 is positive

Elastic potential energy Energy stored in an ideal spring. This is a potential energy that is not gravitational in origin. Although this is not one of the fundamental forces in nature, its potential energy can be defined.

Gravitational potential energy plus Elastic potential energy

Motion with elastic potential energy Ex 7.7 Motion with elastic potential energy m Point 1 m Point 2 What is v2?

Motion with elastic potential energy and work done by other forces: similar to Ex 7.7 The object is moving to the right. So the negative value, -0.6 m/s, is not the solution although it is a mathematical solution.

Motion with gravitational, elastic, and friction forces Ex 7.9 Motion with gravitational, elastic, and friction forces m=2000 kg Point 1 Choose Point 1 as y = 0 Point 2

Conservative and nonconservative forces Allows two-way conversion between kinetic and potential energies Conservative force Properties of the work done by a conservative force The potential energy function can be defined. It is reversible. It is independent of the path of the body; depends only on the starting & ending points When the starting & ending points are the same the total work is zero. f I II i

Nonconservative force Does not allow two-way conversion between kinetic and potential energies The work done by a nonconservative force cannot be represented by a potential energy. Under the influence of some nonconservative force, the body looses its energy. So this is also called a dissipative force. Under the influence of some nonconservative force, the body gets its energy. Potential energy cannot be defined for nonconservative forces!

Conservative or nonconservative? Ex 7.11 Conservative or nonconservative? Leg 3 Leg 4 Leg 2 Leg 1

Force and potential energy integrate Force Potential energy differentiate For 3-dim case For 1-dim case partial derivative

Energy diagram: a graph of energy vs position Energy diagrams Energy diagram: a graph of energy vs position It contains the shape of the potential energy and the total energy is a straight horizontal line as it is a constant once a body is given an energy. This allows to know the motion of the body even if the functional form for the potential energy is not known. Slope of the tangent line is positive, so the force is negative. The body is moving always toward the minimal potential energy point. Slope of the tangent line is negative, so the force is positive.

maximum of the potential: unstable equilibrium point Turning point Minimum of the potential: stable equilibrium point