Dynamics, Work and Energy: Forces, Momentum and Energy

Formulas (2)

• Mass and weight are different quantities · Weight
  $$W = mg$$
• Newton's second law: resultant force equals mass times acceleration · Newton's second law
  $$F_{\text{net}} = ma$$

Reference tables (1)

Watch out for (5)

• The kilogram is not a unit of weight
  → Mass and weight are different quantities
• Motion does not need a resultant force
  → Newton's first law: balanced forces mean constant velocity
• Subtract the weight for anything pushed upwards
  → Newton's second law: resultant force equals mass times acceleration
• At the top of a vertical throw, the weight still acts
  → Newton's second law: resultant force equals mass times acceleration
• Weight and the normal force are not a third-law pair
  → Newton's third law: action and reaction pairs

Formulas (4)

• Static and kinetic friction · Friction
  $$f_s \le \mu_s N \qquad f_k = \mu_k N$$
• Forces on an inclined plane · Sliding down a slope
  $$N = mg\cos\theta \qquad ma = mg\sin\theta - \mu_k mg\cos\theta$$
• Centripetal force for motion in a circle · Centripetal force
  $$F = \frac{mv^2}{r} = m\omega^2 r$$
• Newton's law of universal gravitation · Gravitational force and field strength
  $$F = \frac{G m_1 m_2}{r^2} \qquad g = \frac{GM}{r^2}$$

Watch out for (4)

• Static friction is only as big as it needs to be
  → Static and kinetic friction
• On a slope the normal reaction is less than the weight
  → Forces on an inclined plane
• Uniform circular motion has a non-zero resultant force
  → Centripetal force for motion in a circle
• Measure r from the centre, not from the surface
  → Newton's law of universal gravitation

Formulas (2)

• Momentum and impulse · Momentum and impulse
  $$p = mv \qquad F\,\Delta t = \Delta p = mv - mu$$
• Conservation of momentum in collisions and explosions · Conservation of momentum
  $$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$$

Watch out for (3)

• A rebound changes the momentum by more than a stop does
  → Momentum and impulse
• Momentum is always conserved; kinetic energy only in elastic collisions
  → Conservation of momentum in collisions and explosions
• Velocities after a collision carry signs
  → Conservation of momentum in collisions and explosions

Formulas (3)

• Work done by a force · Work
  $$W = Fs\cos\theta$$
• Kinetic and gravitational potential energy, and the work-energy theorem · Kinetic and potential energy
  $$E_k = \tfrac{1}{2}mv^2 \qquad \Delta E_p = mg\,\Delta h \qquad W_{\text{net}} = \Delta E_k$$
• Hooke's law and elastic potential energy · Hooke's law and stored energy
  $$F = kx \qquad E_{\text{el}} = \tfrac{1}{2}kx^2$$

Watch out for (3)

• No movement along the force means no work
  → Work done by a force
• Kinetic energy goes with the square of the speed
  → Kinetic and gravitational potential energy, and the work-energy theorem
• Stored energy is half force times extension, not force times extension
  → Hooke's law and elastic potential energy

Formulas (2)

• Conservation of mechanical energy, and what friction does · Mechanical energy without friction
  $$\tfrac{1}{2}mv^2 + mgh = \text{constant} \qquad v = \sqrt{2gh}\ \text{(from rest)}$$
• Power and efficiency · Power and efficiency
  $$P = \frac{W}{t} = Fv \qquad \eta = \frac{\text{useful output}}{\text{total input}} \times 100\%$$

Watch out for (2)

• Friction lowers mechanical energy but not total energy
  → Conservation of mechanical energy, and what friction does
• The kilowatt-hour is energy, not power
  → Power and efficiency

Formulas (3)

• The moment of a force, and couples · Moment and couple
  $$M = F d_{\perp} \qquad \tau_{\text{couple}} = F d$$
• Equilibrium and the principle of moments · Principle of moments
  $$\sum M_{\text{clockwise}} = \sum M_{\text{anticlockwise}} \qquad \sum F_{\text{up}} = \sum F_{\text{down}}$$
• Centre of mass and stability · Centre of mass of two masses
  $$x_{\text{cm}} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}$$

Watch out for (3)

• A couple has zero resultant force but a non-zero torque
  → The moment of a force, and couples
• Include the beam's own weight, at its centre
  → Equilibrium and the principle of moments
• The centre of mass is nearer the heavier mass
  → Centre of mass and stability