The Work/Energy Relationship

Work done by a constant net force: Work is the use of force to move an object in the direction of the force. If an object is not moved when a force is applied to it, then no work has been done on that object. Work done by a constant net force: W = Fxd For forces applied at an angle, θ: W=Fcosθ d “d” is the magnitude of displacement

The unit for work is N*m, or more commonly, the Joule. Work is a scalar quantity that is sometimes referred to as being negative.

Work is negative when the force is opposite the displacement. Positive work results in a displacement in the same direction as the applied force. Work is negative when the force is opposite the displacement. Fnet Displacement Friction is doing negative work on the box. Fk Fnet Displacement

Work done by kinetic friction: Frictional work is done whenever the force of kinetic friction hinders motion. For instance, when pulling a sled over concrete, a significant amount of friction prevents the sled from being pulled as far. Work done by kinetic friction: Wf = Fkd

Applying force at an angle can alter the frictional force. If the normal force changes, then so does the force of friction. Fy Fy Reduction of frictional force because Fy is upward, and reduces the normal force. Increase of frictional force because Fy is downward, and increases the normal force.

Net Work Done on an Object: Kinetic Energy of an Object: The net work done on an object can be expressed as the change of the object’s kinetic energy. This expression is known as the “Work - Kinetic Energy Theorem.” Kinetic Energy is the energy of an object due to its motion. Net Work Done on an Object: Wnet = ∆KE = ½mv2 – ½mv02 Kinetic Energy of an Object: KE = ½mv2

Friction is an example of a nonconservative force – one that randomly disperses the energy of the objects on which it acts. For example, the car shown is undergoing frictional forces as it slides. The energy is being dissipated as sound waves, radiant and thermal energy.

Gravity is an example of a conservative force Gravity is an example of a conservative force. It does not dissipate energy. In order to reach the top of the cliff, the man had to use energy to work against gravity. This energy used as work is recovered as KE by diving. Upon reaching the water, his speed gives him kinetic energy equal to the work he used to climb upward.

In general, a force is conservative if the work it does moving an object between two points is the same regardless of the path taken.

Conservative forces can recover the work they do as potential energy. Potential energy is the stored energy that results from an object’s position or condition. It depends only on the beginning and ending points of motion…not the path taken.

Gravitational Potential Energy: Gravitational PE is the product of an objects mass, gravitational acceleration, and height. The work done by gravity is the change in gravitational potential energy. Gravitational Potential Energy: ΔUg = mgΔy *Can also be expressed as “weight x height” Work done by gravity: Wg = ΔUg = ( mgyf – mgyi )

Work done by a nonconservative force: Suppose that friction (or some other nonconnservative force) does work on a mechanical system: For example – a firefighter sliding down a pole to reach the first floor. Work done by a nonconservative force: Wnc = ΔKE + Δ PE

The Law of Conservation of Energy states: ENERGY CAN NEVER BE CREATED OR DESTROYED. Applying this to KE and PE, we develop the Law of Conservation of Mechanical Energy: The total mechanical energy of an isolated system will remain constant. Conservation of Mechanical Energy: Ki + Ui = Kf + Uf If gravity is the only force doing work: ½mv2i + mgyi = ½mv2f + mgyf

Springs , Systems, and Power Chapter 5: Work & Energy Springs , Systems, and Power

Spring force is another conservative force. When you stretch or compress a spring with an applied force, the work you did can be recovered by removing that force. Springs store the work done on them as Elastic PE. A spring is said to be at Equilibrium when it is neither stretched nor compressed. “x” represents the displacement from equilibrium.

By applying a force to a spring, you displace it from equilibrium. In the example, a mass is hung from the spring, and it’s weight stretches the spring. Doubling the mass doubles the distance stretched. The weight of the mass, mg, stretches the spring some distance, x, from equilibrium.

Hooke’s Law Fspring = -kx So, we can say that the distance from equilibrium is directly proportional to the force applied. This relationship is known as Hooke’s Law. “k” represents the spring constant. The force is also known as the restoring force because the spring always exerts an equal and opposite force that will restore it to equilibrium. Hooke’s Law Fspring = -kx

Elastic Potential Energy When a spring is stretched or compressed, it holds elastic potential energy. This is due to that “restoring force” that will return the spring to equilibrium. Elastic Potential Energy PEs = ½kx2

Conservation of Mechanical Energy Now, by adding in PEs to the conservation of energy, we get: Conservation of Mechanical Energy (K+ Ug + Us)i = (K+ Ug + Us)f

Work done by NC Forces: Wnc = Ef - Ei When studying systems objects, the underlying concept is: The work done on a system by nonconservative forces equals the change in mechanical energy. Where “E” represents Energy. Work done by NC Forces: Wnc = Ef - Ei

Average Power Delivered: Power is the rate at which energy is transferred over time. The unit for power is the Watt (W)…equivalent to one Joule per second. The US Unit is Horsepower: 1 HP = 746 W Average Power Delivered:

Average Power Delivered: Power can also be expressed in terms of Force and Velocity: Work = Fd. So… Recall that speed = d/t. So… Average Power Delivered:

Another measure of energy is the kilowatt-hour. 1 kWh is the energy transferred in one hour at the rate of 1000 Joules per Second. So, 1 kWh is equivalent to 3.6 x 106 Joules of energy.