6. Further Mechanics

• To keep an object moving in a circle at a constant speed, a constant centripetal force is required (a force applied always towards the centre of that circle)
• An object moving in a circle at a constant speed is accelerating (True). The direction is always changing, hence the velocity always changing, where acceleration is defined as the change in velocity over time
• Equations to calculate the magnitude of angular speed ( ⍵ ):

  • ⍵ = v/r
  • ⍵ = 2 𝝅 ƒ

  ⍵ - s^-1

  v - ms^-1

  r - m

  f - Hz
• Angular acceleration in terms of angular velocity:

  A = ⍵²r

  ⍵ - s^-1

  a - ms^-2

  r - m
• Angular acceleration in terms of velocity:

  a = v² / r

  a - ms^-2

  v - ms^-1

  r - m
• Equations for centripetal force:

  • F = mv²/r
  • F = m ⍵²r

  ⍵ - s^-1

  v - ms^-1

  r - m

  F - N

  m - kg
• A radian is the angle of a circle sector such that the radius is equal to the arc length. Radians are usually written in terms of 𝝅 (i.e., 2𝝅 radians = 360 degrees)
• Conditions for Simple Harmonic Motion (SHM):

  • Acceleration must be proportional to its displacement from the equilibrium point
  • It must act towards the equilibrium point
  • a ∝ - ��
• Constant of proportionality linking acceleration and 𝑥:

  • ⍵² or - k/m
• 𝑥 as a trig function of t and ⍵:

  • 𝑥 = Acos( ⍵ t) or 𝑥 = Asin( ⍵ t)

  ⍵ - s^-1

  A - m

  x - m

  t - s
• Calculating the maximum speed using ⍵ and A:

  Max speed = ⍵ A

  ⍵ - s^-1

  A - m
• Calculating the maximum acceleration using ⍵ and A:

  Max acceleration = ⍵²A

  ⍵ - s^-1

  A - m
• Equation for the time period of a mass-spring simple harmonic system:

  T = 2 𝜋 √(m/k)

  m - kg

  k - Nm^-1

  T - s
• Equation for the time period of a simple harmonic pendulum:

  T = 2 𝜋 √(l/g)

  g - ms^-2

  l - m

  T - s
• Small angle approximation for sin 𝑥:

  sin 𝑥 ≈ 𝑥

  Valid in radians
• Small angle approximation for cos 𝑥:

  cos 𝑥 ≈ 1 - 𝑥²/2

  Valid in radians
• Definition of Free Vibrations:

  The frequency a system tends to vibrate at in free vibration is called the natural frequency
• Definition of Forced Vibrations:

  A driving force causes the system to vibrate at a different frequency. For resonance, the phase difference will be π/2 radians
• Definition of damping and types:

  • Damping occurs when an opposing force dissipates energy to the surroundings
  • Critical damping reduces the amplitude to zero in the quickest time
  • Overdamping is when the damping force is too strong and returns to equilibrium slowly without oscillation
  • Underdamping is when the damping force is too weak and it oscillates with exponentially decreasing amplitude
• Effect of greater damping on vibration:

  For a vibration with greater damping, the amplitude is lower at all frequencies due to greater energy losses from the system. The resonant peak is also broader because of the damping
• Implications of resonance in real life:

  Soldiers must break stop when crossing bridges and vehicles must be designed so there are no unwanted vibrations
• Explain what is meant by resonance.


  • Idea that (at resonance) frequency of forced vibrations equals natural/resonant frequency
  • Idea that amplitude (of vibrations/oscillations) is at a maximum
• A ship floating in the sea can be modelled by the pencil floating in water. The ship can oscillate vertically. These oscillations are called heave oscillations.

  Wave motion causes forced oscillations of the ship. Under certain conditions, heave resonance may then occur.

  (c) Explain what is meant by resonance.

  • Idea that (at resonance) frequency of forced vibrations equals natural/resonant frequency
  • Idea that amplitude (of vibrations/oscillations) is at a maximum
• As the spacecraft rotates, a force that imitates the effect of gravity acts on an astronaut who is in contact with the floor. Explain why.

  • Centripetal force acts inwards / towards the centre of rotation
  • Links reaction force to centripetal force

  
• In this spacecraft mA < mB.

  Deduce whether the centre of mass of A or the centre of mass of B rotates with a greater linear speed.

  • The angular speed is the same for A & B or
  • Rotational radius for B less than that for A
  • Both of these points AND v = rω so velocity of A is greater
• Describe, with reference to one of Newton’s laws of motion, the evidence that a resultant force is acting on the block.

  • Block constantly changing direction (at constant speed)
  • Uses N1 (every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force)to show that a force must apply
• Suggest the effect this has on the amplitude relationship and the phase relationship between the moving shadows.

  • Amplitude – the pendulum shadow amplitude becomes less than the block shadow amplitude
  • Phase – time period decreases therefore phase changes

  
• Discuss the consequences for the forces acting on the pole when one acrobat has a much greater mass than the other.

  • Vertical (compressive) force on the pole increases
  • Increases mass increases weight and hence tension in the rope (for the same angle)
  • Centripetal Force
  • on the acrobats/masses would be different/not equal
  • Unbalanced moments acting (on pole)/resultant torque acting (on pole)
  • Causing the pole to sway/bend/move/ or tilt/topple the platform toward more massive acrobat
• Explain two differences in the graph for spring B.

  • (resonance) peak / maximum amplitude is at a higher frequency
  • due to higher spring constant
  • (resonant) peak would be broader
  • due to damping

  
• State the conditions for simple harmonic motion.

  SHM is when:

  • The acceleration is proportional to the displacement
  • the acceleration is in opposite direction to displacement
• Explain how and why this happens.

  • when the vibrating surface accelerates down with an acceleration less than the acceleration of free fall the sand stays in contact.
  • above a particular frequency, the acceleration is greater than g
  • there is no contact force on the sand
  • sand no longer in contact when downwards acceleration of plate is greater than acceleration of sand due to gravity

  
• • Time a minimum of 10 oscillations in total including at least one repeat the measurement & calculate the mean period of oscillation, T, of the bat
  • Measure the distance, l, from the pivot to the sweet spot
  • Calculate the period, Tc, for a simple pendulum of length l using Tc = 2π√(l / 9.81)
  • Calculate the uncertainty in T from the repeat measurements
  • The %uncertainty in Tc = 0.5 × the %uncertainty in l, if Tc was calculated
  • Compare T with Tc to see whether they agree within the calculated uncertainties.
  • An experimental detail designed to reduce uncertainties (but not more repeat readings)

  
• Discuss the motion of the ball in terms of the forces that act on it.

  • First law: ball does not travel in a straight line, so a force must be acting on it
  • although the ball has a constant speed its velocity is not constant because its direction changes constantly
  • because its velocity is changing, it is accelerating
  • Second law: the force on the ball causes the ball to accelerate in the direction of the force
  • the acceleration (or change in momentum) is in the same direction as the force
  • the force is centripetal: it acts towards the centre of the circle

  
• Discuss the motion of the ball in terms of the forces that act on it.

  • Third law: the ball must pull on the central point of support with a force that is equal and opposite to the force pulling on the ball from the centre
  • the force acting on the point of support acts outwards
  • the ball is supported because the rope is not horizontal
  • there is no resultant force in the vertical direction
  • the weight of the ball, mg, is supported by the vertical component of the tension, F cos θ
  • the horizontal component of the tension, F sin θ, provides the centripetal force