LECTURE 17 POTENTIAL ENERGY & CONSERVATIVE FORCES

Instructor: Kazumi Tolich Lecture 17

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¨ Reading chapter 8-1 to 8-2 ¤ Conservative and non-conservative forces ¤ Potential energy

Conservative forces

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The net work done by a conservative y force on a particle moving around any closed path is zero. rf The work done by a conservative force on a particle moving between two points r does not depend on the path taken by i the particle. x ¨ Force of gravity and a force exerted by a massless spring are both conservative Recall work forces. W = Fd cosθ Gravity is a conservative force

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¨ As the box moves in a closed path, gravity does zero work. Non-conservative forces

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¨ Any force that is not conservative force is a non- conservative force. ¤ Kinetic friction ¤ Tension in a rope ¤ Forces exerted by a motor ¤ Forces exerted by muscles ¤ etc Kinetic friction is a non-conservative force

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¨ The total work done by the kinetic friction on the box is not zero as the box moves in a closed path. Example: 1

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a) Calculate the work done by gravity as a 5.2-kg object is moved from A to B in the figure along paths 1 and 2.

b) How do your results depend on the mass of the block? Specifically, if you increase the mass, does the work done by gravity increase, decrease, or stay the same? Potential energy

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¨ Potential energy (U) is energy stored in the configuration of a physical system. ¤ A single point-like object cannot have potential energy since it cannot have any configuration.

¨ Potential energy can be converted into other forms of energy, such as kinetic energy, and do work in the process.

¨ Potential energy is a scalar quantity.

¨ Potential energy is measured in joules, J. Different types of potential energy

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¨ Gravitational potential energy ¤ Stored in a configuration of at least two gravitationally attracting objects. n e.g. Earth and a book, the sun and Earth, etc

¨ Elastic potential energy ¤ Stored in a deformed spring, or a similar object.

¨ Electric potential energy (future quarter) ¤ Stored in a configuration of at least two electrically interacting charged objects.

¨ etc Potential energy functions

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¨ The work done by a conservative force on a particle does not depend on the path, but the endpoints of the path.

¨ We define the potential-energy function U that is associated with a conservative force as

ΔU = U f − Ui = −Wc

where Wc is the work done by the internal conservative force. e.g. A falling ball-Earth system loses its potential energy as the gravitational force does work on the ball by pulling it toward Earth. Gravitational Ug function 11

¨ A system of an object and Earth can store gravitational potential energy.

¨ Change in gravitational potential energy depends only on the

difference in height, (yf – yi).

ΔUg = mg(yf − yi ) ¨ You can arbitrarily choose a reference height where gravitational potential energy of a system is at a reference value.

¨ Only the difference in gravitational potential energy that is independent of your choice of reference point is the meaningful quantity.

¨ If we define Ug = 0 at y = 0, Ug = mgy. External work and Ug 12

¨ Consider a system of an object with mass m and Earth. A constant external force F is lifting the object a height of y without any acceleration.

¨ The magnitude of the external force must equal to that of the gravitational force (internal to the system) on the object.

¨ The work done by the external force on the object is

Wext = Fy = (mg)y = ΔUg

¨ The external work done on the system changes the gravitational potential energy if the kinetic energy of the object does not change. External work and Ug: 2 13

¨ Consider now a system consists of an object with mass m only, and excludes Earth. A constant external force F is applied to lift the object a height of y without any acceleration.

¨ The net force on the object is zero.

¨ The work done by the external force on the object is

Wext = (F − w)y = (mg − mg)y = 0 = ΔUg

¨ There is not net work done on the object by the external force, and the gravitational potential energy of the system (just the object) does not change. Demo: 1

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¨ Generator Driven by a Falling Weight ¤ Demonstration of gravitational potential energy converted into electric energy through work. Clicker question: 1 Example: 2

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¨ As a cliff diver drops to the water from a height of 40.0 m, the gravitational potential energy of the man- Earth system decreases by 25,000 J. How much does the diver weigh? Example: 3

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¨ At time t = 0.00 s, a box with mass m = 2.00 kg is dropped from a

height of h0 = 20.0 m above the ground. Neglect air resistance. a) What is the potential energy of the box-Earth system relative to the ground at t = 0.00 s? b) Find the height of the box and its speed at t = 1.00 s. c) Find the potential energy of the box-Earth system and the kinetic energy of the box at t = 1.00 s. d) Find the kinetic energy and the speed of the box just as it reaches the ground. Clicker question: 2

18 Elastic Us function 19

¨ A system of an object attached to a deformed spring stores elastic potential energy.

¨ Choosing Us to be 0 at x = 0, when the spring is not compressed or stretched, the elastic potential energy is given by

1 2 Us = kx 2 x = 0

¨ Elastic potential energy stored in a particular state of the system is equal to the external work required to achieve that state. Example: 4

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¨ A spring with a spring constant of 3200 N/m is initially stretched until the elastic potential energy is 1.44 J

(Us = 0 when it is not stretched or compressed). What is the change in the elastic potential energy if the initial stretch is changed to a) a stretch of 2.0 cm, b) a compression of 2.0 cm, and c) a compression of 4.0 cm? Clicker question: 3

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