work energy and power(Pant sir notes)

Front

Back

What is the definition of work in physics?

Work is said to be done when a force applied on a body displaces the body through a certain distance in the direction of the force.

What is the formula for work done by a constant force?

W = F s cos θ = F⃗ · s⃗

When is work done by a force positive?

When the force (or its component) is parallel to the displacement (0° ≤ θ < 90°). The external force favours the motion.

When is work done by a force negative?

When the force (or its component) is opposite to the displacement (90° < θ ≤ 180°). The external force opposes the motion.

When is work done by a force zero?

Under three conditions: (1) Force is perpendicular to displacement (θ = 90°), (2) Displacement is zero (s = 0), (3) Force is zero (F = 0).

What is the maximum and minimum work done by a force of magnitude F over displacement s?

W_max = F s (when θ = 0°); W_min = –F s (when θ = 180°).

What is the formula for work done by a variable force?

W = ∫_A^B F⃗ · d s⃗ = ∫ (F cos θ) ds

How is work done calculated from a force-displacement graph?

Work done = Area under the force-displacement curve (with proper algebraic sign).

What are the absolute units of work?

SI: 1 Joule = 1 N · 1 m; CGS: 1 erg = 1 dyne · 1 cm. Relation: 1 Joule = 10⁷ erg.

What are the gravitational units of work?

SI: 1 kg-m = 9.81 Joule; CGS: 1 gm-cm = 981 erg.

What is a conservative force/field?

A force for which work done is independent of the path and work done over a closed path is zero. Examples: gravitational, electrostatic, elastic forces.

What is a non-conservative force?

A force for which work done depends on the path and work done over a closed path is not zero. Examples: friction, viscous force, air drag.

What is the definition of energy?

Energy of a body is its capacity for doing work. It is a scalar quantity with the same dimensions as work [ML²T⁻²].

What is the SI unit of energy and important practical units?

SI: Joule. Practical: 1 eV = 1.6 × 10⁻¹⁹ J, 1 kWh = 3.6 × 10⁶ J, 1 calorie = 4.18 J.

What is Einstein’s mass-energy equivalence?

E = mc². 1 amu ≡ 931 MeV; 1 kg ≡ 9 × 10¹⁶ J.

What is kinetic energy and its expression?

Energy possessed by a body by virtue of its motion. KE = ½ mv².

What is the work-energy theorem?

Work done by a force acting on a body equals the change in its kinetic energy: W = ΔK = ½ m(v² – u²). Valid for all types of forces.

What is the relation between kinetic energy and linear momentum?

KE = P² / (2m) or P = √(2m KE). A body cannot have KE without momentum and vice-versa.

What is the stopping distance of a vehicle under a constant retarding force F?

x = (½ mv²) / F = KE / F.

What is the stopping time of a vehicle under a constant retarding force F?

t = (mv) / F = P / F.

What is potential energy?

Energy associated with the position or configuration of a body in a conservative field. Defined only for conservative forces.

How is change in potential energy related to work by a conservative force?

U₂ – U₁ = –∫ F⃗ · d r⃗ = –W_conservative.

What is the three-dimensional formula relating force and potential energy?

F⃗ = –∇U (force equals the negative gradient of potential energy).

What are the conditions for stable, unstable and neutral equilibrium from the potential-energy curve?

Stable: dU/dx = 0 and d²U/dx² > 0 (U minimum). Unstable: dU/dx = 0 and d²U/dx² < 0 (U maximum). Neutral: dU/dx = 0 and d²U/dx² = 0 (U constant).

What is elastic potential energy stored in a spring?

U = ½ k x² = ½ F x = F² / (2k), where k is the spring constant.

What is the spring constant k?

k = F / x. Numerically equal to the force required to produce unit extension or compression. Dimension [MT⁻²], SI unit N/m.

What is gravitational potential energy of a mass m at height h above the Earth’s surface (h ≪ R)?

U = mgh.

What is the exact gravitational potential energy at height h above Earth’s surface?

U = mgh / (1 + h/R), where R is Earth’s radius.

What is the work done in pulling a chain of length L and mass M with (1/n)th of its length hanging over a frictionless table?

W = MgL / (2 n²).

What is the velocity of a chain of length L when it completely leaves a frictionless table after (1/n)th was initially hanging?

v = √[g L (1 – 1/n²)].

State the law of conservation of mechanical energy.

In the presence of only conservative forces, the sum of kinetic and potential energies of an isolated system remains constant: K + U = constant.

What happens to mechanical energy when non-conservative forces (e.g., friction) are present?

Δ(K + U) = W_friction (or –Q if heat is produced). Total energy (including heat, etc.) is still conserved.

What is the definition of power?

Power is the rate at which work is done (or energy is transferred). Average power P_av = W / t; Instantaneous power P = F⃗ · v⃗.

What are the SI unit and practical units of power?

SI: Watt (1 W = 1 J/s). Practical: 1 hp = 746 W, 1 kW = 10³ W, 1 MW = 10⁶ W.

What is the dimension of power?

[ML²T⁻³].

How is work related to a power-time graph?

Work done = Area under the power-time curve.

For an automobile of mass m starting from rest under constant power P, how do velocity and position vary with time?

v = (2 P t / m)^{1/2}; s = (8 P / 9 m)^{1/2} t^{3/2}.

What is a collision?

An isolated event in which a strong force acts between two or more bodies for a short time, changing their energy and momentum.

What is always conserved in any collision?

Total linear momentum and total energy (including all forms). Kinetic energy is conserved only in perfectly elastic collisions.

Define the coefficient of restitution e.

e = (relative velocity of separation) / (relative velocity of approach) = (v₂ – v₁) / (u₁ – u₂). Dimensionless; 0 ≤ e ≤ 1.

What are the values of e for different types of collision?

Perfectly elastic: e = 1; Perfectly inelastic: e = 0; Inelastic: 0 < e < 1.

What are the final velocities after a one-dimensional perfectly elastic collision of two masses m₁ and m₂?

v₁ = [(m₁ – m₂)/(m₁ + m₂)] u₁ + [2 m₂ /(m₁ + m₂)] u₂; v₂ = [(m₂ – m₁)/(m₁ + m₂)] u₂ + [2 m₁ /(m₁ + m₂)] u₁.

What happens when two bodies of equal mass undergo a head-on elastic collision?

Their velocities are interchanged.

What happens when a light projectile collides elastically head-on with a stationary heavy target?

The projectile rebounds with almost the same speed; the target remains almost at rest.

What is the fractional loss of kinetic energy of the projectile in a head-on elastic collision with a stationary target?

ΔK / K = 4 m₁ m₂ / (m₁ + m₂)². Maximum (100 %) when m₁ = m₂.

In a perfectly elastic oblique collision of two equal masses (target at rest), what is the angle between the final velocities?

θ + φ = 90° (the two bodies move at right angles to each other after collision).

What are the final velocities after a one-dimensional inelastic collision with coefficient of restitution e?

v₁ = [(m₁ – e m₂)/(m₁ + m₂)] u₁ + [(1 + e) m₂ /(m₁ + m₂)] u₂; v₂ = [(1 + e) m₁ /(m₁ + m₂)] u₁ + [(m₂ – e m₁)/(m₁ + m₂)] u₂.

What is the loss of kinetic energy in a one-dimensional inelastic collision?

ΔK = ½ [m₁ m₂ /(m₁ + m₂)] (1 – e²) (u₁ – u₂)².

What is the common velocity after a perfectly inelastic collision of two masses moving in the same direction?

v_comb = (m₁ u₁ + m₂ u₂) / (m₁ + m₂).

What is the loss of kinetic energy in a perfectly inelastic collision?

ΔK = ½ [m₁ m₂ /(m₁ + m₂)] (u₁ – u₂)².

If a ball is dropped from height h₀ onto a floor with coefficient of restitution e, what is the height after the first rebound?

h₁ = e² h₀.

What is the height of a bouncing ball after the nth rebound?

hₙ = e^{2n} h₀.

What is the total distance travelled by a bouncing ball before it stops?

H = h₀ (1 + e²) / (1 – e²).

What is the total time taken by a bouncing ball before it stops?

T = [(1 + e)/(1 – e)] √(2 h₀ / g).

In a ballistic-pendulum type collision (bullet embeds in a suspended block), what is the velocity of the combined system just after collision?

v = m u / (m + M).

How is the initial velocity of the bullet related to the height h reached by the combined system?

u = [(m + M)/m] √(2 g h).

What is the loss of kinetic energy when a bullet of mass m and speed u embeds in a block of mass M?

ΔK = ½ [m M /(m + M)] u².