Two Chapter 2

Energy Transformation in a Pendulum

• To move the pendulum to A, work is done which gives the pendulum potential energy Ep = -mgh
• At A (release position) kinetic energy Ek = 0 because V = 0
• When at B, Ek > 0 because V > 0 and Ep has decreased because h has decreased
• At C, h = 0 (the pendulum is at the reference level) therefore Ep = 0 and Ek is a maximum because V = a maximum.
• As it moves to D, Ek decreases because V decreases and Ep increases because h increases
• Under all circumstances, because energy is conserved we have

E_p + E_k = E_m = a constant

• E_m = mechanical energy
• Note that in this description, we have ignored friction.

Elastic Potential Energy and Hooke's Law

• Some objects when deformed, return to their original shape when allowed.
• Elastic potential energy can be stored in these types of objects.
  • E.g.
    • diving boards
    • trampolines
    • rubber bands
    • sling shot
    • springs
• Hooke's Law - The amount of deformation of an elastic object is proportional to the force applied to deform the object.
• For springs, the graph of Force needed to stretch or compress a spring versus the amount of stretch or compression (x) is

• Elastic limit - the greatest stretch or compression of an object that shows a linear relation between the force and the amount of stretch or compression.
  • beyond this limit, objects may not return to their natural shape.
  • The equation for the linear portion of the graph is

F = kx

• F = force, k is a proportionality constant, and x is the amount of stretch or compression.
• the constant k, has units of N/m
• Both F and x are negative if compression occurs.
• The direction of the force and the direction of deformation are always the same

Elastic Potential Energy Stored in a Spring

• Recall that the work done on an object is equal to the area under the force versus displacement graph.
• Consider a spring which has a force F applied to it to stretch it distance x

• Recall that the units of k are N/m therefore substituting units only into the equation

Ee = (1/2)kx² gives

Ee = (1/2)(N/m)(m)² = N.m = J

• Therefore we conclude that the units of Ee are the Joule.

Gravitational Potential Energy Near The Earth's Surface

• We calculated the elastic potential energy for a spring by calculating area under the Force versus displacement graph. This is the same thing as calculating work done.
• We can calculate gravitational potential energy in the same way.

W = F.Dd and

F = -mg (the force to lift an object ) therefore

W = -mgDh (Dh is the change in height or the distance the force is applied over)

Therefore, as stated before:

DE_g = - mgDh

• DE_g = the change ingravitational potential energy.
• This equation only holds true for small values of Dh compared to the radius of the Earth.
  • This is true because g changes as height changes
• Points to remember when using DE_g = - mg Dh
  1. DE_g = change in gravitational E_p
  2. Dh must be small so g does not vary
  3. Dh is vertical displacement from Earth's center
  4. DE_g is + when moving away from the Earth and negative when moving towards Earth.
  • Because of these limitations on DE_g = -mgDh, we need to look at a more general circumstance that allows for g to change.

Gravitational Potential Energy in General

Combining Potential and Kinetic Energy

• An object in orbit, has both kinetic and potential energy.
• Similarly, an object that is launched into orbit has large kinetic energy near the Earth's surface, and this kinetic energy gets transformed into potential energy as it rises up to orbit. (Vise Vera when the object falls)
• Total energy is always conserved (first law of thermodynamics), therefore

E_t = E_k + E_p = E_m

• Mechanical Energy - The sum of kinetic and potential energy
  • is conserved if the system is conservative
  • i.e.

E_m = E_k + E_p = a constant (for a conservative system)

• Conservative System - One in which friction does not act, or is so small it can be ignored.
• If friction is significant, then mechanical energy is not conserved, but total energy is conserved.

E_t = E_k + E_p + Q + Light + ....... = a constant

• E_t = Total energy, Q = heat, and the ..... refers to all other forms of energy involved.

December 6, 2013