Laws of Motion — JEE Mains Physics

Formulas (3)

• Second law as a vector and along a rope · Second law
  $$\vec F_{\text{net}} = m\vec a, \qquad T = m(g \pm a)$$
• Impulse and change of momentum · Impulse–momentum
  $$F_{\text{avg}}\,\Delta t = \Delta p, \qquad |\Delta p|_{\text{rebound}} = m(v + v')$$
• Variable mass: rockets, guns, belts and jets · Force from a mass flow
  $$F = v_{\text{rel}}\frac{dm}{dt}, \qquad P_{\text{belt}} = v^{2}\frac{dm}{dt}$$

Watch out for (8)

• Steady climbing needs only mg
  → Second law as a vector and along a rope
• The upthrust stays the same when ballast is dropped
  → Second law as a vector and along a rope
• A removed force leaves its opposite
  → Second law as a vector and along a rope
• A rebound doubles the change
  → Impulse and change of momentum
• Kinetic energy is not conserved when mass is added
  → Impulse and change of momentum
• Belt power is v² dm/dt, not half of it
  → Variable mass: rockets, guns, belts and jets
• A jet's force has v twice
  → Variable mass: rockets, guns, belts and jets
• A rocket's thrust must also lift its weight
  → Variable mass: rockets, guns, belts and jets

Formulas (2)

• Strings, chains and forces at a point · Equilibrium at a point
  $$\sum F_x = 0,\quad \sum F_y = 0, \qquad T_0 = \frac{mg}{2}\cot\theta$$
• Normal reactions, smooth inclines and supports · Normal reaction
  $$N = mg \pm F\sin\theta, \qquad F_{\text{horizontal}} = mg\tan\theta$$

Watch out for (6)

• The lower half of the rope carries only the weight
  → Strings, chains and forces at a point
• The lowest-point tension is not half the weight
  → Strings, chains and forces at a point
• The steeper string carries more
  → Strings, chains and forces at a point
• N is not always mg
  → Normal reactions, smooth inclines and supports
• Constant velocity means zero net force
  → Normal reactions, smooth inclines and supports
• A smooth contact pushes perpendicular to the surface
  → Normal reactions, smooth inclines and supports

Formulas (3)

• One acceleration, then one tension at a time · Atwood machine
  $$a = \frac{(m_2 - m_1)g}{m_1 + m_2}, \qquad T = \frac{2m_1m_2\,g}{m_1 + m_2}$$
• Movable pulleys and string constraints · Constraint
  $$\sum T_i\,a_i = 0, \qquad a_{\text{load}} = \frac{a_{\text{end}}}{2}$$
• Friction inside a pulley system · Table block and hanging block
  $$a = \frac{m_h g - \mu_k m_t g}{m_h + m_t}$$

Watch out for (7)

• The tension is not the hanging weight
  → One acceleration, then one tension at a time
• Divide by the total mass
  → One acceleration, then one tension at a time
• Read which ratio is asked
  → One acceleration, then one tension at a time
• Equal accelerations do not hold with a movable pulley
  → Movable pulleys and string constraints
• The movable pulley feels 2T
  → Movable pulleys and string constraints
• Check that it moves before using kinetic friction
  → Friction inside a pulley system
• No sliding, no friction
  → Friction inside a pulley system

Formulas (3)

• Friction on a level surface · Friction and the least starting pull
  $$f_s \le \mu_s N,\quad f_k = \mu_k N, \qquad F_{\min} = \frac{\mu mg}{\cos\theta + \mu\sin\theta}$$
• Holding and pushing a block on a rough incline · Push up and hold
  $$F_{\text{up}} = mg(\sin\theta + \mu\cos\theta), \qquad F_{\text{hold}} = mg(\sin\theta - \mu\cos\theta)$$
• Sliding down a rough incline · Sliding down
  $$a = g(\sin\theta - \mu\cos\theta), \qquad \mu = \tan\theta\left(1 - \frac{1}{n^{2}}\right)$$

Watch out for (9)

• N is not mg when the force is at an angle
  → Friction on a level surface
• Static friction is only as large as it needs to be
  → Friction on a level surface
• Area of contact does not matter
  → Friction on a level surface
• Friction flips direction between push and hold
  → Holding and pushing a block on a rough incline
• The contact force is the full weight
  → Holding and pushing a block on a rough incline
• μ > tan θ means it will not slide by itself
  → Holding and pushing a block on a rough incline
• Time ratios are squared
  → Sliding down a rough incline
• Friction on a slope is μ mg cos θ
  → Sliding down a rough incline
• An extra force can change N
  → Sliding down a rough incline

Formulas (2)

• Pseudo force in an accelerating frame · Pseudo force
  $$\vec F_{\text{pseudo}} = -m\vec a_0, \qquad \tan\theta = \frac{a_0}{g}$$
• Forces in circular motion · Banked road with friction
  $$v_{\max}^{2} = rg\,\frac{\tan\theta + \mu}{1 - \mu\tan\theta}$$

Reference tables (1)

Watch out for (7)

• Velocity does not decide the reading
  → Apparent weight in a lift
• A lift at constant velocity changes nothing
  → Apparent weight in a lift
• The pseudo force points against the acceleration
  → Pseudo force in an accelerating frame
• Use it in one frame only
  → Pseudo force in an accelerating frame
• μ = v²/(rg) only on a flat road
  → Forces in circular motion
• Convert km/h before squaring
  → Forces in circular motion
• Centrifugal force belongs to the rotating frame
  → Forces in circular motion