Conservation of Energy - II

Conservation of Energy - II

6.4 Gravitational potential energy (1)

• Stored energy due to the interaction of an object with something else that can easily be recovered as kinetic energy is called potential energy (symbol U).

• Since W

=F

Δy=-mgΔy,

• Forces that have potential energies associated with them are called conservative forces.

– Not every force has associated potential energy: ex) kinetic friction

• The sum of the kinetic and potential energies (K+U) is called the mechanical energy (E

).

• W is equal to the change in mechanical energy.

• Only changes in potential energy enter our calculations.

– In W_nc=ΔK+ ΔU, only the change in potential energy enters the calculation.

– Most often, we choose some convenient position and assign it to have zero potential energy.

– There is no special significance to the sign of the potential energy. What matters is the sign of the potential energy change.

• Rcognizing a conservative force

– If the force is conservative, the work done by the force is independent of path.

– Energy stored as potential energy by a conservative force during a displacement from A to point B can be recovered as kinetic energy.

– The work done by friction, air resistance, and other contact forces does depend on path, so these forces cannot have potential energies associated with them.

6.5 Gravitational potential energy (2)

• When the approximation for gravitational force, i.e, F=mg, is not valid, the potential turns out to be as follows:

• For a very small displacement from r_i to r_f=r_i+Δy,

y mg r y

m GM

y r

r

r y m r

y GM r

m r GM

r m GM y

r

m U GM

U U

E

i i

i i

E i

i E

i E i

E i

f

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











 



 





 

 





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) (

) ) (

1 (1

) (

) (

2

• The choice (U=0 when r=∞) means that the gravitational potential energy is negative for any finite value r.

– The sign of U has no particular significance.

6.6 Work done by variable forces: Hooke’s law

• To calculate the work done by a variable force F

, we divide the overall displacement into a series of small diplacements Δx.

• The total work done is the area under the graph of F

(x)

from x

to x

.

• Hooke’s law and ideal springs

– Hooke’s law : the deformation of the object is

proportional to the magnitude of the force that causes the deformation

– Ideal spring : massless and follows Hooke’s law.

– The constant k is called the spring constant, and its SI units are N/m.

• Work done by an ideal sping

– The work done by spring as its end moves from 0 to x;

– More generally, if the moveable end starts at position xi, not necessarily at the equilibrium, the work done by the spring is

6.7 Elastic potential energy

• The kind of potential energy stored in a spring is called elastic potential energy.

• Just as for gravity, the change in elastic potential energy is the negative of the work done by spring:

• The most convenient choice is to assign U=0 when the

spring is relaxed (x=0):

6.8 Power

• We give the name power (symbol P) to the rate of energy transfer.

• The average power is the amount of energy converted (ΔE) divided by the time transfer takes (Δt):

• The SI unit of power is given the name watt (1 W= 1 J/s).

– The kilowatt-hour (kW·h) is a unit of energy.

• The work done by a force during a small time interval Δt is W=FΔr·cosθ.

• Since Δr=vΔt, ^{cos cos Fvcos}

t F r t

r F t

P W 



 



 

 