Work and EnergyWork and Energy

A A calculuscalculus-based perspective-based perspectiveAP Physics CAP Physics C

The old “special case”The old “special case” W=F dW=F d is a special case is a special case

F must be in the direction F must be in the direction of motionof motion

F is a constant forceF is a constant force What if F is not parallel What if F is not parallel

to d?to d? The Dot or Scalar Product.The Dot or Scalar Product. One vector times another One vector times another

to “make” a scalarto “make” a scalar W= FcosW= Fcosθθdd

If the graph below shows the force exerted by the Death If the graph below shows the force exerted by the Death Star’s Tractor beam. How could you calculate the work Star’s Tractor beam. How could you calculate the work

done on a ship being pulled from position a to b?done on a ship being pulled from position a to b?

A.A. Find the slope of Find the slope of the line between a the line between a and band b

B.B. Find the Area Find the Area under the curve under the curve from point a to bfrom point a to b

C.C. Multiply force x (b-Multiply force x (b-a)a)

Area Under the CurveArea Under the Curve For Hooke’s Law, the force is linear.For Hooke’s Law, the force is linear. For a force vs. distance graph, For a force vs. distance graph, the the

work is just the area under the work is just the area under the curvecurve. The shape is a triangle.. The shape is a triangle.

What if the area under the curve is What if the area under the curve is not geometric?not geometric?

•Finding an integral is a way of accurately finding the area under a curve.

•My looking at smaller and smaller “pieces” of x, multiplying them by the force, and summing them we can find total area.

•Most Forces aren’t constant.

•Springs•Magnetic fields•Pushes/pulls•Gravity over large distances

The RealityThe Reality

Integration Rules!!Integration Rules!!

Example: Work done to stretch Example: Work done to stretch a spring.a spring.

F(x) = kxF(x) = kx Hooke’s LawHooke’s Law

How much work do you do if you How much work do you do if you stretch a spring from an initial stretch a spring from an initial

position of 0m to the 1m position if position of 0m to the 1m position if the spring constant is 49 N/mthe spring constant is 49 N/m

A.A. 49 Nm49 NmB.B. 25 Nm25 NmC.C. 98 Nm98 NmD.D. 10 Nm10 Nm

Work for a varying forceWork for a varying force2

1

( )x

x

Work F x dx

cosb b

aba a

W F ds F ds

A force F(x) acts on a particle. The force is related to the position of the particle by the formula F(x) = Cx3, where C is a

constant. Find the work done by this force on the particle when the particle moves from x = 1.5 m to x = 3 m.

A.A. 1 1 JJB.B. 34 34 JJC.C. 0 0 JJD.D. 19C 19C JJ

SolutionSolution

3 33

1.5 1.5

( )W F x dx Cx dx

34

4 4

1.5

3 1.5 194 4x CW C C J

EnergyEnergy

Work-Energy TheoremWork-Energy Theorem The change in the kinetic energy of The change in the kinetic energy of

an object is equal to the net work an object is equal to the net work done on the object.done on the object.

Types of ForceTypes of Force ConservativeConservative

Obeys conservation Obeys conservation of energyof energy

ExamplesExamples Spring forceSpring force Gravity Gravity

Non-conservativeNon-conservative

Energy is Energy is transferred into transferred into non-mechanical non-mechanical formsforms

ExamplesExamples FrictionFriction Air dragAir drag

EquilibriumEquilibrium Occurs when net force = 0Occurs when net force = 0 If force If force = F(x),= F(x), then equilibrium exists at then equilibrium exists at

points where points where F(x) =F(x) = 0. 0.

Equilibrium exists where Equilibrium exists where dU/dxdU/dx = 0 = 0

( )U F x dx

PowerPower

W Fd madP Fvt t t

Power takes many formsPower takes many forms