• The?total energy?of an object oscillating with?SHM?is the?sum?of?its?potential energy?(gravitational?or?elastic) and?kinetic energy

E = E_P?+ E_K

• Where:
  • E?= total energy in joules (J)
  • E_P?= potential energy in joules (J)
  • E_K?= kinetic energy in joules (J)

  

Graph of total energy E, potential energy E_P?and kinetic energy E_K?of an object oscillating with SHM

• If energy losses due to friction or drag are zero or ignored, the?total energy?E??of the system is?conserved?
• The?potential energy?store of the object is at a?maximum?at the point of maximum displacement from the equilibrium position
  • The point of maximum displacement is?amplitude?x₀
  • Kinetic energy is zero at amplitude
  • Potential energy is equal to the total energy of the system at this point

• Energy is?transferred?from the object's potential energy store to its kinetic energy store as the object moves from amplitude to the equilibrium position
  • The object has both potential and kinetic energy
  • The sum of the potential and kinetic energy is equal to the total energy of the system
  • The total energy of the system is conserved

• The?kinetic energy?store of the object is at a?maximum?at the?equilibrium position
  • This is because velocity is at a maximum as the object passes through the equilibrium position
  • Kinetic energy is equal to the total energy of the system at this point

• Energy is?transferred?from the object's kinetic energy store to its potential energy store as the object moves from the equilibrium position to amplitude
  • The object has both potential and kinetic energy
  • The sum of the potential and kinetic energy is equal to the total energy of the system
  • The total energy of the system is conserved

(i) Determine the total energy of the object by reading the maximum value of the kinetic energy from the graph?

• • From the graph, read the maximum value of the object's kinetic energy?(E_K)_MAX?= 60 mJ
  • Recall that, at the equilibrium position (x?= 0), the total energy?E?is exactly equal to the maximum value of the kinetic energy?(E_K)_MAX
  • Since energy losses can be neglected, the total energy is constant

E?= 60 mJ

(ii) Read the amplitude of the object's oscillations from the graph?

• • The maximum displacement positions are the locations on either side of the equilibrium position where the kinetic energy is zero?E_K?= 0

x₀?= 2.0 cm

(iii)

Step 1: Recall the equation for the kinetic energy?E_K?of an object in terms of its mass?m?and velocity?v

Step 2: Rearrange the above equation to calculate the velocity?v

Step 3: Substitute the numbers into the equation to calculate the maximum velocity of the object?

• • Mass of the object,?m?= 0.50 kg
  • You must convert the maximum kinetic energy must from millijoules (mJ) into joules (J)

E_K?= 60 mJ = 0.06 J

v?= 0.49 m s^–1

(iv)

Step 1: Read the value of the kinetic energy?E_K?of the object when the displacement is?x?= 1.0 cm

E_K?= 50 mJ

Step 2: Write down the relationship between total energy E, kinetic energy?E_K?and potential energy?E_P

E = E_P?+ E_K

Step 3: Rearrange the above equation to calculate the potential energy?E_P

E_P?= E –? E_K

Step 4: Substitute the numbers in the above equation?

• • E_K?= 50 mJ
  • E?= 60 mJ

E_P?= 60 mJ – 50 mJ

E_P?= 10 mJ