General Physics I Work & Energy

General Physics I Work & Energy

Forms of Energy Kinetic: Energy of motion. A car on the highway has kinetic energy. We have to remove this energy to stop it. The brakes of a car get HOT! This is an example of turning one form of energy (motion) into another (thermal energy).

Energy Conservation Energy cannot be destroyed or created. Just changed from one form to another. We say energy is conserved! True for any closed system. When we put on the brakes, the kinetic energy of the car is turned into heat using friction in the brakes. The total energy of the car-brakes-road-atmosphere system is the same. The energy of the car alone is not conserved... It is reduced by the braking. Doing work on an isolated system will change its energy!

Definition of Work: Ingredients: Force (F), displacement ( x) Work, W, of a constant force F acting through a displacement x is: W = F x = F x cos = F x x Units: Force x Distance = Work F F x x Newton x Meter = Joule

Non-Conservative Work If the work done does not depend on the path taken, the force is said to be conservative. If the work done does depend on the path taken, the force is said to be non-conservative. The most common example of a non-conservative force is friction. Thus, the work done by friction in pushing an object a distance D is given by: W f = -f D NOTE: The work done by friction is ALWAYS negative (opposite the direction of motion)

Work on an Incline A large red box is pulled up a rough (m > 0) incline by a rope-pulley-weight arrangement as shown below. How many forces are doing work on the red box? (a) (b) 3 (c) 4 W TOT = F TOT x It s the total force that matters!! Is work done by N?

Work & Kinetic Energy A force F pushes a box across a frictionless floor for a distance x. The speed of the box is v o before the push and v f after the push. Since the force F is constant, acceleration a will be constant. We have shown that for constant a: v f - v o = a(x f -x o ) = a x (one of our original equations of motion). Multiply by 1 / m: 1 / mv f - 1 / mv o = ma x But F = ma 1 / mv f - 1 / mv o = F x = W F v o F v f m a x x

Work & Kinetic Energy... So we found that: 1 / mv f - 1 / mv o = F* x = W F Define Kinetic Energy, KE: KE = 1 / mv KE f - KE i = W F W F = KE (Work/kinetic energy theorem) {Net Work done on object} = {Change in kinetic energy of object}

Work & Energy Two blocks have masses m 1 and m, where m 1 > m. They are sliding on a frictionless floor and have the same kinetic energy when they encounter a long rough stretch (i.e. m > 0) which slows them down to a stop. Which one will go farther before stopping? HINT: Think about the relation that: KE = W F = F x (a) m 1 (b) m (c) They will go the same distance m 1 m

A simple application: Work done by gravity on a falling object What is the speed of an object after falling a distance H, assuming it starts at rest? W net KE v 0 = 0 H

Potential Energy For any conservative force we can define a potential energy function PE g such that: PE g = PE gf PE gi = -W net mgy f mgy i The potential energy function PE g is always defined only up to an additive constant (PE g = mgh) The potential energy is based on the position of an object rather than any motion (kinetic energy) that it may have. You can choose the location where PE g = 0 to be anywhere convenient.

More about the Conservation of Energy If only conservative forces are present, the total kinetic plus potential energy of a system is conserved, i.e. the total mechanical energy is conserved. (note: E = E mechanical throughout this discussion) E = KE + PE E = KE + PE = W + PE = W + (-W) = 0 using KE = W using PE = -W E = KE + PE is constant!!! Both KE and PE can change, but E = KE + PE remains constant. But we ve seen that if non-conservative forces act then energy can be dissipated into other modes (thermal, sound, etc.) Power is the rate of doing work : P W t

Falling Objects Three objects of mass m begin at height h with velocity 0. One falls straight down, one slides down a frictionless inclined plane, and one swings on the end of a pendulum. What is the relationship between their velocities when they have fallen to height 0? v=0 v=0 v=0 H v f v i v p Free Fall Frictionless incline Pendulum (a) V f > V i > V p (b) V f > V p > V i (c) V f = V p = V i

Example: The simple pendulum Suppose we release a mass m from rest a distance h 1 above its lowest possible point. What is the maximum speed of the mass and where does this happen? To what height h does it rise on the other side? m h 1 h v

Problem: Hotwheel A toy car slides on the frictionless track shown below. It starts at rest, drops a distance d, moves horizontally at speed v 1, rises a distance h, and ends up moving horizontally with speed v. Find v 1 and v. d v 1 v h

Potential Energy & The Spring For a spring we know that F x = -kx. F(x) x o x f W s 1 k x f x o x relaxed position -kx Change in the potential energy of a spring (PE s ) F = - k x o F = - k x f

Conservation of Energy & The Spring A block slides on a horizontal frictionless surface with a speed v. It is brought to rest when it hits a bumper that compresses a spring. How much is the spring compressed? An object is released from rest at a height H on a curved frictionless ramp. At the foot of the ramp is a spring of constant k. The object slides down the ramp and into the spring, compressing it a distance x before coming momentarily to rest. What is x?

End of Work & Energy Lecture