Work, Energy and Power — JEE Mains Physics

Formulas (3)

• Work done by a constant force · Work by a constant force
  $$W = \vec F\cdot\vec s = Fs\cos\theta = F_x s_x + F_y s_y + F_z s_z$$
• Work done by a force that varies with position · Work by a variable force
  $$W = \int_{x_1}^{x_2} F(x)\,dx \qquad W = \int F_x\,dx + \int F_y\,dy$$
• Slowing down under a retarding force that depends on position · Energy after a position-dependent retarding force
  $$\tfrac12 mv^{2} = \tfrac12 mu^{2} - \int_{a}^{b}|F(x)|\,dx$$

Watch out for (6)

• A distance along a direction needs the unit vector
  → Work done by a constant force
• A force that balances friction does positive work
  → Work done by a constant force
• Area below the axis is negative work
  → Work done by a force that varies with position
• A force like x²y î depends on the path
  → Work done by a force that varies with position
• Use the patch's end points, not its length
  → Slowing down under a retarding force that depends on position
• A retarding force takes energy away
  → Slowing down under a retarding force that depends on position

Formulas (3)

• Work-energy theorem with several forces · Work-energy theorem
  $$W_{\text{net}} = \sum W_i = K_f - K_i$$
• Work from a given velocity or position law · Work from two speeds
  $$W_{\text{net}} = \tfrac12 m\left(v_2^{2} - v_1^{2}\right), \quad v = \frac{dx}{dt}$$
• Kinetic energy and momentum, K = p²/2m · Kinetic energy and momentum
  $$K = \frac{p^{2}}{2m} \qquad p = \sqrt{2mK}$$

Watch out for (6)

• Gravity also works through the depth of penetration
  → Work-energy theorem with several forces
• Constant speed does not mean every force does zero work
  → Work-energy theorem with several forces
• Do not assume the body starts from rest
  → Work from a given velocity or position law
• Work done on the body, not by it
  → Work from a given velocity or position law
• Square the factor, not the percentage
  → Kinetic energy and momentum, K = p²/2m
• A percentage increase subtracts the starting value
  → Kinetic energy and momentum, K = p²/2m

Formulas (3)

• Conservative forces and potential energy · Force from potential energy
  $$F = -\frac{dU}{dx} \qquad F_x = -\frac{\partial U}{\partial x} \qquad W_{\text{cons}} = -\Delta U$$
• Potential energy of a spring · Spring potential energy
  $$U = \tfrac12 kx^{2} \qquad W_{x_1 \to x_2} = \tfrac12 k\left(x_2^{2} - x_1^{2}\right)$$
• Conservation of mechanical energy · Mechanical energy conserved
  $$\tfrac12 mv_1^{2} + mgh_1 = \tfrac12 mv_2^{2} + mgh_2 \qquad v_{\text{bottom}}^{2} = 5gL$$

Watch out for (9)

• Work is the integral of F·dr; the minus sign belongs to ΔU
  → Conservative forces and potential energy
• The force is the slope, not the height
  → Conservative forces and potential energy
• Friction has no potential energy
  → Conservative forces and potential energy
• Spring energy is half of force times stretch
  → Potential energy of a spring
• A dropped ball falls the extra compression too
  → Potential energy of a spring
• Energy between two stretches uses x₂² − x₁²
  → Potential energy of a spring
• A rod is not a string
  → Conservation of mechanical energy
• Measure every height from one level
  → Conservation of mechanical energy
• Both masses on a pulley carry kinetic energy
  → Conservation of mechanical energy

Formulas (2)

• Power from force and velocity · Power
  $$P = \vec F\cdot\vec v \qquad P_{\text{avg}} = \frac{W}{t} \qquad \vec v = \frac{1}{m}\int_{0}^{t}\vec F\,dt$$
• Motion under constant power · Constant power from rest
  $$v = \sqrt{\frac{2P}{m}}\,t^{1/2} \qquad x = \sqrt{\frac{8P}{9m}}\,t^{3/2}$$

Watch out for (5)

• Instantaneous power needs the velocity at that instant
  → Power from force and velocity
• Count friction and efficiency
  → Power from force and velocity
• Per hour means per 3600 seconds
  → Power from force and velocity
• Constant power is not constant force
  → Motion under constant power
• The power is t^(3/2), not t^(2/3)
  → Motion under constant power

Formulas (3)

• Impulse and change in momentum · Impulse
  $$\vec J = \int \vec F\,dt = \Delta\vec p \qquad F_{\text{avg}} = \frac{\Delta p}{\Delta t}$$
• Conservation of momentum in explosions, recoil and sticking collisions · Momentum conserved
  $$m_1\vec u_1 + m_2\vec u_2 = m_1\vec v_1 + m_2\vec v_2 \qquad \Delta K_{\text{lost}} = \frac12\frac{m_1m_2}{m_1 + m_2}\left(u_1 - u_2\right)^{2}$$
• Bullet and pendulum: momentum in the impact, energy in the swing · Ballistic pendulum
  $$mu = (M + m)V \qquad V = \sqrt{2gh} \qquad V_{\text{full circle}} = \sqrt{5gL}$$

Watch out for (8)

• A rebound adds the two speeds
  → Impulse and change in momentum
• At an angle, only the normal part reverses
  → Impulse and change in momentum
• Same impulse, smaller force
  → Impulse and change in momentum
• Momenta add as vectors
  → Conservation of momentum in explosions, recoil and sticking collisions
• Kinetic energy is not conserved when bodies stick
  → Conservation of momentum in explosions, recoil and sticking collisions
• An explosion can increase kinetic energy
  → Conservation of momentum in explosions, recoil and sticking collisions
• Do not conserve energy through the impact
  → Bullet and pendulum: momentum in the impact, energy in the swing
• A bullet that bounces back gives the bob extra momentum
  → Bullet and pendulum: momentum in the impact, energy in the swing

Formulas (2)

• Head-on elastic collisions · Elastic collision with a target at rest
  $$v_1 = \frac{m_1 - m_2}{m_1 + m_2}\,u \qquad v_2 = \frac{2m_1}{m_1 + m_2}\,u \qquad \frac{K_2}{K_1} = \frac{4m_1m_2}{(m_1 + m_2)^{2}}$$
• Coefficient of restitution and bouncing balls · Coefficient of restitution
  $$e = \frac{v_2 - v_1}{u_1 - u_2} \qquad h' = e^{2}h \qquad \frac{K'}{K} = e^{2}$$

Watch out for (5)

• Exchange of velocities happens one collision at a time
  → Head-on elastic collisions
• Maximum compression is at the common velocity
  → Head-on elastic collisions
• A light body bounces back from a heavy one
  → Head-on elastic collisions
• Height goes with e², speed with e
  → Coefficient of restitution and bouncing balls
• Every bounce is travelled twice
  → Coefficient of restitution and bouncing balls