Work and Energy

Outcomes Upon completion of this unit you will be able to: Analyze force problems in terms of energy. Define the term "work" as it relates to physics. Calculate work problems using the various equations. Identify which forces are at work in a given problem situation. Interpret a force displacement curve in relation to work. Calculate work as done by a spring. Define power and efficiency. Calculate power and efficiency using the appropriate formulas. Demonstrate an understanding that power is a product of force and velocity.

Objectives Con’t Demonstrate an understanding of kinetic energy as a concept and as an equation. Calculate the change in kinetic energy using the work-energy theorem formula. Explain potential energy. Calculate gravitational potential energy. Define spring force. Calculate spring force using the formula for Hooke's Law. Describe conservative forces. Write the conservative energy statement using the correct mathematical forms. Solve energy problems using the appropriate energy equations.

Work Work is done if a force is applied for a particular distance. In Physics there is one more stipulation: the force and displacement vectors must point in the same direction. W = Fxdx Fx is the applied force in the direction of the object's displacement and dx is the object's displacement. Why the x-subscript? This shows that in order for work to be done, the force and displacement must be in the same direction.

Extreme Work This formula for work leads us to three possible extreme cases Force and Displacement in the Same Direction : (results in max. positive work) Example?

Force Perpendicular to Displacement No work is done in this case Example?

Force and Displacement in Opposite Directions this results in max. negative work Example?

Identify the Force at Work It is extremely important to identify what force is doing the work in question. Look at the case of lifting then lowering a box.

Lifting a box As you lift the box, you exert a force in the same direction as the displacement, so you do positive work. Gravity (weight) always acts straight down and here that means it is in the opposite direction to the displacement, so it does negative work.

Lowering the box As the box is lowered, you still exert an upward force to keep the box from simply falling. Now your force and the displacement are at 180 degrees, so you do negative work. Gravity is now doing positive work.

Units for Work The unit of work is the joule (J). lifting a good sized apple at a constant velocity straight up for a distance of 1 m requires about 1 J of work. While work can be positive or negative and its calculation depends on direction, it is not a vector. Work is a scalar. Direction is important for calculating work, but work itself has no direction.

Try It You exert a force of 20 N in order to slide a textbook across a table a a constant speed. If the textbook has a mass of 20 kg and you slide it a distance of 50 cm, How much work do you perform?

Try it Again A 1500 kg car is brought to a complete stop over a distance of 43 m. If the coefficient of friction between the car and the road is 0.32, how much work is done by friction in bringing the car to a stop? Start by drawing a rough FBD of the situation:

Force Displacement Graph Another useful fact is that the area under a force displacement curve is equal to the work done. Of course, the force that is plotted will have to be the component that is in the same direction as the displacement.

Work done on a Spring The work done on a spring when it is compressed or stretched is given by the formula shown below: where k is the spring constant (N/m) and x is the amount of compression (m). You can not use the basic work formula W = Fx for a spring. The formula W = Fx assumes that the force is constant. As you well know, the force a spring exerts (F) changes as the displacement (x) changes. You must use the special formula above if you need to calculate the work done by a spring.

Power and Efficiency Power is simply the rate at which you do work. The formula for average power is shown below. ΔW = change in work (J) Δt= change in time (s) The unit of power is the joule/second or the watt (W). When you see a light bulb rated at 60 W, it means that it consumes 60 J of electrical energy every second.

Example An elevator motor lifts a mass of 1000 kg over a distance of 20 m in 15 seconds. What power must it develop?

A winch rated at 1.5 kW pulls a heavy box along a horizontal floor. It takes the winch 1.00 min to pull the box over a distance of 250 m. What Force is it exerting?

Power as a product of F and V It is possible to show that power is equal to the force times the velocity. The formula for this is shown below: P = Fv F = applied force (N) v = object's velocity (m/s)

Efficiency Efficiency is the ratio of energy output to energy input: Efficiency = E o /E i x 100% E o – energy output E i – energy input It can also be written as a ratio of work output to work input: Efficiency = W o /W i x 100% W o – work output W i – work input

Example A toaster transforms 1200 J of electrical energy into 560 J of thermal energy to make a piece of toast. What is the efficiency of the toaster? We know that: Ei = 1200 J Eo = 560 J So our efficiency is: