6. Work, energy and power

Work, Energy, and Power

  • Designed by Edustudy Point

Scalar Product

  • The scalar product (dot product) of vectors A and B is represented as A.B.

  • Formula: A.B = AB Cosθ, where θ is the angle between the vectors.

  • A, B, and cos θ are scalars; hence, the dot product is a scalar quantity without direction.

  • B represents the product of the magnitude of A and the component of B along A, and vice versa.

  • Properties:

    • Commutative: A.B = B.A

    • Distributive: A.(B + C) = A.B + A.C

Notions of Work and Kinetic Energy

  • Work-Energy Theorem: Change in kinetic energy (Kf - Ki) is equal to work done (W) by the net force.

  • Formula: Kf - Ki = W

  • In 3D motion: v^2 - u^2 = 2a.d (where u = initial velocity, v = final velocity, a = acceleration, d = displacement).

  • Using this, we have: ½ mv^2 - ½ mu^2 = ma.d = F.d (Here, ma = F).

Work

  • Work done (W) is defined as the product of the component of force (F) in the direction of displacement (d).

  • Formula: W = F.d

  • Conditions for No Work:

    • Displacement is zero.

    • Force is zero.

    • Force and displacement are perpendicular.

  • Variable Force:

    • Variable forces are more common than constant forces. For small displacement (Dx), assume F(x) is approximately constant: DW = F(x) Dx.

Kinetic Energy (K)

  • Kinetic energy is a measure of the work an object can do due to its motion.

  • Formula: K = (1/2)mv^2

  • Kinetic energy is a scalar quantity.

Potential Energy (P)

  • Potential energy is the stored energy due to position/configuration: P = mgh.

  • If force F(x) can be expressed mathematically, potential energy can also be defined based on conservative forces.

Conservative and Non-Conservative Forces

  • Conservative Forces: Work done depends only on initial and final positions. Examples: Gravitational force, Electrostatic force.

  • Non-Conservative Forces: Work done depends on velocity or path taken. Example: Frictional force.

Conservation of Mechanical Energy

  • Mechanical energy associates with motion and position of an object, represented by K + V(x).

  • For conservative forces, ΔK = ΔW = F(x) Δx, leading to Δ(K + V) = 0.

  • Kinetic (K) and Potential Energy (V(x)) may vary but their sum remains constant.

Conservative Force Definition

  • A force is conservative if derived from a scalar quantity V(x): F(x) = -dV/dx.

  • Work done depends only on end points.

Potential Energy of a Spring

  • The spring force (Fs = -kx) follows Hooke’s law, where k is the spring constant (N/m).

  • The energy stored in a spring: V(x) = (1/2)kx^2, defined as zero when at equilibrium.

Various Forms of Energy

  • Heat: Work done by friction is not lost but converts into heat energy.

Chemical Energy

  • Arises from binding energies in molecules.

  • Exothermic Reactions: Release heat (example: freezing water).

  • Endothermic Reactions: Absorb heat (example: melting ice).

Electrical Energy

  • Associated with the flow of electric current (lights, fans, etc.).

Nuclear Energy

  • Released from nuclear reactions (fission or fusion).

  • Relates to mass-energy equivalence: E = mc².

  • Mass defect ( ∆m) in nuclear reactions can be significant.

Principle of Conservation of Energy

  • Non-conservative forces can transform mechanical energy into heat, light, or sound.

  • The total energy of the universe remains constant.

Power

  • Defined as the rate at which work is done or energy is transferred.

  • Average Power: Pav = W/t.

  • Instantaneous Power: P = dW/dt.

  • Formulas: W = F.dr, P = F.(dr/dt), P = F.v (where v is instantaneous velocity).

  • Power is a scalar quantity, SI unit: Watt (W). 1 hp = 746 W.

Collisions

  • Defined as events where two or more bodies exert forces on each other for a short time.

  • Total linear momentum is conserved in all collisions.

Types of Collisions

  • Elastic Collisions: Initial KE equals final KE.

  • Inelastic Collisions: Some KE is lost.

  • Completely Inelastic Collisions: Bodies stick together post-collision.

One-Dimensional Collisions

  • When initial and final velocities are along the same straight line.

Two-Dimensional Collisions

  • Illustrated involving multiple directions (ongoing examples provided).

Elastic Collision Example

  • Involving two masses, m1 and m2. Initial and final momentum and kinetic energy equations are useful for analysis.

  • Special Cases in Elastic Collision defined based on mass ratios and velocities.