Spring Potential Energy

• Work done by an applied force is stretching or compressing a spring can be recovered by removing the applied force, so, like gravity, the spring force is conservative.

• In fig. a) spring in equilibrium position (the spring is not compressed or stretched)

b) pushing the block against the spring, compresses it a distance x ( displacement from the equilibrium when x=0)

• Experimentally, doubling a given displacement, requires double the force, while tripling it takes triple force

• Hook’s Law: Fs = -kx

Fs- force exerted by spring

k- spring constant, is a constant of proportionality

SI unit: k = N/m

Fs is called a restoring force, because the spring always exerts a force in a direction opposite the displacement of its end, tending to restore whatever is attached to the spring to its original position

• Elastic potential energy, can be associated with the spring force and is equal to the negative of the work done by the spring

• The average force: F = (F0+F1)/2 = (0-kx)/2 = -kx/2The work done by the spring force: Ws= F x = -1/2kx2

In general the spring is compressed or stretched form xi to xf:

Ws = - (1/2k xf2 – 1/2k xi

2)

• The Work –Energy Theorem:

Wnc- (1/2k xf2 – 1/2k xi

2) = ΔKE +ΔPEg

PEg is gravitational potential energy

the Potential Energy of spring:

PEs= ½ kx2

Wnc = (KEf-KEi) +(PEgf – PFgi) +(PEsf-PEsi)

Wnc – work done by nonconservative forces

KE – kinetic energy

PEs – elastic potential energy

Peg – gravitational potential energy

c) When the block is released, the spring snaps back to its original length, and the elastic potential energy is covered by Kinetic Energy of the block

In the absents of neoconservative forces, Wnc = 0:

(KEf-KEi) +(PEgf – PFgi) +(PEsf-PEsi)=0

(KE + PEg+ PEs)i = (KE + PEg+ PEs)f

System and energy conservation• The work energy theorem:

W nc+ Wc = ΔKE

Wc= - ΔPE

Wnc = ΔKE + ΔPE

=(KEf- KEi) + (PEf- PEi)

= (KEf +PEf) –(KEi + PEi)

Total mechanical energy :

E = KE =PE

Wnc = Ef- Ei= ΔE

• If the mechanical energy is changing, it has to be going somewhere.

• The energy either leaves the system and goes into the surrounding environment, or it stays in the system and is converted into a nonmechanical form such as thermal energy

• Principle of conservation of energy: Energy is conserved, it can’t be created or destroyed, only transfer from one form into another

• POWER

If an external force is applied to an object and if the work done by this force is W in the time interval Δt, then the average power delivered to the object during this interval is the work done divided by the time interval:

P = W /t

P =W /t =FΔx/ Δt = F v

Si unit: watt ; W= J/s = 1kg m2/s3

• 1 hp = 550 ft lb/s = 746 W

• One kilowatt –hour (kWh) is hthe energy transfer in one hour at the constant rate of 1kW =1000J/s

1kW = (103W)3600 = 3.6 x106J

WORK DONE BY A VARYING FORCE:

Suppose an object is displaced along the x-axis under the action of a force Fx.

The object is displaced in the direction of increasing x from x=xi to x=xf

W1 =Fx Δx

W = F1 Δx1 + F2 Δx2 + …. + Fn Δxn

The work done by a variable force acting on an object that undergoes a displacement is equal to the area under the graph of Fx versus x