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PHYS 218 sec Review Chap. 7 Potential Energy and Energy Conservation

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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 y1y1 y2y2 When the body moves up, the work done by the gravitational force is negative and the potential energy increases.

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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.

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

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Ex 7.1Height 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.

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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

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

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

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

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

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

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

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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.

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Gravitational potential energy plus Elastic potential energy

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

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Ex 7.8 Motion with elastic potential energy and work done by other forces: similar to Ex 7.7 m m 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.

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

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Conservative and nonconservative forces Conservative force Allows two-way conversion between kinetic and potential energies 1.The potential energy function can be defined. 2.It is reversible. 3.It is independent of the path of the body; depends only on the starting & ending points 4.When the starting & ending points are the same the total work is zero. Properties of the work done by a conservative force I II i f

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Nonconservative force Does not allow two-way conversion between kinetic and potential energies 1.The work done by a nonconservative force cannot be represented by a potential energy. 2.Under the influence of some nonconservative force, the body looses its energy. So this is also called a dissipative force. 3.Under the influence of some nonconservative force, the body gets its energy. Potential energy cannot be defined for nonconservative forces!

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

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

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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. Slope of the tangent line is negative, so the force is positive. The body is moving always toward the minimal potential energy point.

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

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