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Mechanics
Work, Energy, and Power
Work is defined as the scalar product of force and displacement vectors, representing the transfer of mechanical energy. Energy is the scalar capacity of a physical system to perform work, while power measures the rate of doing work or transferring energy.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Calculation of work by constant and variable forces, kinetic and potential energy theorems, conservation of mechanical energy, and basic power problems.
PGT extension
Conservative and non-conservative force fields, line integrals of force, potential energy curves, stability of equilibrium, and work-energy relations in rotating frames and variable mass systems.
Prerequisites
Newton's laws of motion and free-body diagrams
Vector algebra (dot product and line integrals)
Kinematics of rectilinear and curvilinear motion
Work Done by Constant and Variable Forces
Work W is defined mathematically as the line integral of force F along a displacement path C: $W = \int F \cdot dr$.
For a constant force, this simplifies to $W = Fs \cos(\theta)$, where θ is the angle between the force and displacement vectors.
Work can be positive (θ < 90°), negative (90° < θ ≤ 180°), or zero ($\theta = 90°)$.
Conservative Forces and Potential Energy
A force is conservative if the work done by it along a closed path is zero, or equivalently, if the work done is independent of the path taken.
For a conservative force field, $F = -\nabla U$, where U is the potential energy function.
Non-conservative forces, such as kinetic friction and air resistance, dissipate mechanical energy into thermal energy.
Work-Energy Theorem
The net work done on a particle by all forces equals the change in its kinetic energy: Wnet = Δ$K = K_{f} - K_{i}$.
This theorem is valid in both inertial and non-inertial frames provided pseudo forces are included in the net force calculation.
It provides a first integral of the equation of motion, simplifying problems where time is not explicitly required.
Power and Efficiency
Instantaneous power P is defined as the scalar product of force and velocity: $P = F \cdot v$.
Average power is the total work done divided by the total time interval: Pavg = ΔW / Δt.
Mechanical efficiency η is the ratio of useful output power to total input power.
Derivations
Work-Energy Theorem for Rectilinear Motion
Conditions: Particle of mass m moving under the action of a net force F along a straight line.
Start with Newton's second law: $F = m (dv/dt)$.
Express acceleration using the chain rule: dv/d$t = (dv/dx)(dx/dt) = v (dv/dx)$.
Substitute into the force equation: $F = mv (dv/dx)$, which rearranges to F dx = m v dv.
Integrate both sides from initial position xi (velocity vi) to final position xf (velocity vf): ∫ F d$x = \int mv dv$.
Relation Between Conservative Force and Potential Energy
Conditions: 1D conservative force field where work done depends only on endpoints.
Define the change in potential energy ΔU as the negative of the work done by the conservative force: Δ$U = -W = -\int _{xi}^{x_{f}} F(x) dx$.
Differentiate both sides with respect to the final position xf using the Fundamental Theorem of Calculus.
Obtain dU/d$x = -F(x)$.
Rearrange to express force in terms of potential energy: F(x) = -dU/dx.
Result: $F = -\frac{dU}{dx}$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Work done by a constant force
$$W = F \cdot s = Fs \cos\theta$$
F = force vector, s = displacement vector, $\theta = angle between F$ and s Joule (J) or N·m
Force must be constant in magnitude and direction
Work done by a variable force (1D)
$$W = \int_{x_1}^{x_2} F(x) \, dx$$
F(x) = position-dependent force, x1, x2 = initial and final positions Joule (J)
None
Kinetic Energy
$$K = \frac{1}{2}mv^2 = \frac{p^2}{2m}$$
m = mass, v = velocity, p = linear momentum Joule (J)
Valid for speeds v << c (non-relativistic mechanics)
Potential Energy (Gravitational - Uniform Field)
$$U = mgh$$
m = mass, g = acceleration due to gravity, h = height above reference level Joule (J)
Near Earth's surface where g is constant
Potential Energy (Spring)
$$U = \frac{1}{2}kx^2$$
k = spring constant, x = elongation or compression from natural length Joule (J)
Spring must obey Hooke's law
Conservation of Mechanical Energy
$$E = K + U = \text{constant}$$
E = total mechanical energy, K = kinetic energy, U = potential energy Joule (J)
Only conservative forces do work (no non-conservative dissipation)
Instantaneous Power
$$P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$$
W = work, t = time, F = force vector, v = velocity vector Watt (W) or J/s
None
Work-Energy Theorem
$$W_{\text{net}} = \Delta K$$
Wnet = net work done by all forces, ΔK = change in kinetic energy Joule (J)
Valid in inertial frames
Graphs and experiments
Force-Displacement Graph
Draw: A plot of force F on the vertical axis versus displacement x on the horizontal axis for a variable force.
Interpret: The area under the F-x curve represents the total work done by the force during the displacement interval.
Exam use: Tested via numerical interpretation of irregular geometric shapes (rectangles, triangles) under force curves.
Potential Energy Curve (U vs x)
Draw: A curve showing potential energy U as a function of position x, including local minima, maxima, and inflection points.
Interpret: Minima correspond to stable equilibrium (F=0, d2U/dx2 > 0); maxima correspond to unstable equilibrium (d2U/dx2 < 0).
Exam use: Used to determine equilibrium points, turning points of motion, and bound/unbound states.
Work Done by Spring Force (Hooke's Law Experiment)
Draw: Linear plot of restoring force F versus displacement x for a helical spring.
Interpret: The slope of the F-x line yields the spring constant k, and the triangular area gives the stored elastic potential energy.
Exam use: Questions evaluating spring energy storage and elastic potential energy changes.
Solved numericals
TGT core
A block of mass 2 kg is pulled across a frictionless floor by a constant horizontal force of 10 N for a distance of 5 m. Find the work done by the force and the final kinetic energy of the block, assuming it starts from rest.
Identify given values: F = 10 N, s = 5 m, $\theta = 0°$, m = 2 kg, u = 0.
Calculate work done: $W = Fs \cos(0°) = 10 N \times 5 m \times 1 = 50 J$.
Answer: Work done = 50 J, Final Kinetic Energy = 50 J
PGT extension
A conservative force in a 1D system is given by F(x) = -ax + bx2, where a and b are positive constants. Find the potential energy function U(x), taking U = 0 at x = 0, and determine the positions of equilibrium.
Use the relation F(x) = -dU/dx, which gives d$U = -F(x) dx$.
Evaluate the integral: U(x) = \frac{1}{2}ax^2 - \frac{1}{3}bx^3.
For equilibrium, F(x) = 0 ⇒ -ax + bx2 = 0 ⇒ x(bx - a) = 0.
Equilibrium positions are x = 0 and $x = a/b$.
Answer: U(x) = \frac{1}{2}ax^2 - \frac{1}{3}bx^3; Equilibrium at x = 0 and $x = a/b$
PGT extension
An elevator of mass 500 kg is ascending with a constant upward acceleration of 2 m/s2 over a height of 10 m. Calculate the total work done by the tension cable on the elevator. (Take $g = 9.8 m/s^{2})$
Apply Newton's second law for the elevator: T - m$g = ma \Rightarrow T = m(g + a)$.
The displacement s = 10 m upwards, in the same direction as tension.
Calculate work done by tension: WT = T · s = 5900 N × 10 m = 59,000 J.
Answer: 59,000 J (or 59 kJ)
Assumptions and limitations
Classical mechanics assumes macroscopic particles where quantum uncertainty in position and momentum is negligible.
Potential energy is defined only for conservative force fields; non-conservative systems require tracking thermal energy dissipation.
Common misconceptions
Common error: Work is always positive if a force is applied and an object moves.
Correct: Work can be negative if the force component opposes the direction of displacement (θ > 90°).
Work is a scalar product taking into account the relative angle between force and displacement vectors.
Common error: Centripetal force does zero work, so it can change the speed of a particle in circular motion.
Correct: Centripetal force does zero work because it is always perpendicular to instantaneous displacement, hence speed remains constant in uniform circular motion.
$W = Fs \cos(90°) = 0$, so by the work-energy theorem, kinetic energy and speed do not change.
Quick revision
Work is the area under the F-x curve and represents energy transfer.
The work-energy theorem equates net work to change in kinetic energy.
Conservative forces satisfy $F = -dU/dx$ and path independence.
Must remember
Work is defined mathematically as the line integral of force along a displacement path: $W = \int \vec{F} \cdot d\vec{r}$.
For a constant force, $W = Fs \cos\theta$, where $\theta$ is the angle between the force and displacement vectors.
A force is conservative if the work done by it along a closed path is zero or independent of the path.
The relationship between a conservative force field and potential energy is given by $\vec{F} = -\nabla U$.
The net work done on a particle by all forces equals the change in its kinetic energy: $W_{\text{net}} = \Delta K = K_f - K_i$.
Instantaneous power is defined as the scalar product of force and velocity: $P = \vec{F} \cdot \vec{v}$.
Mechanical energy is conserved only when no non-conservative forces like friction do work.
Glossary
Work ($W$)The energy transferred to or from an object via the application of force along a displacement, measured in Joules.
Conservative ForceA force for which work done is independent of path and depends only on initial and final positions.
Work-Energy TheoremStates that the net work done by all forces acting on a body equals its change in kinetic energy.
Power ($P$)The rate at which work is done or energy is transferred, measured in Watts ($J/s$).
Non-conservative ForceA force, such as kinetic friction, that dissipates mechanical energy into non-recoverable thermal energy.
Conservative vs Non-Conservative Forces
Property
Conservative Forces
Non-Conservative Forces
Path Dependence
Independent of path
Dependent on path
Work in Closed Loop
Zero
Non-zero (dissipative)
Associated Potential Energy
Exists ($F = -\nabla U$)
Does not exist
Examples
Gravitational force, Spring force
Friction, Air resistance
Practice Quiz
Choose an option to reveal the explanation.
1. What is the mathematical definition of work done by a variable force along a path $C$?
Explanation: Work is defined fundamentally as the line integral of force along the displacement path: $W = \int \vec{F} \cdot d\vec{r}$.
2. If the angle $\theta$ between the constant force vector and displacement vector is $120^\circ$, the work done is:
Explanation: Since $90^\circ < \theta \le 180^\circ$, $\cos(120^\circ) = -0.5$, making the work done negative.
3. Which of the following forces is a non-conservative force?
Explanation: Kinetic friction dissipates mechanical energy into thermal energy and is a classic example of a non-conservative force.
4. For a conservative force field, the relation between force $\vec{F}$ and potential energy $U$ is:
Explanation: A conservative force is related to potential energy gradient by $\vec{F} = -\nabla U$.
5. What does the Work-Energy Theorem state?
Explanation: The net work done on a particle by all forces equals the change in its kinetic energy: $W_{\text{net}} = \Delta K$.
6. Instantaneous power $P$ can be expressed as the scalar product of force and which other vector?
Explanation: Instantaneous power is defined as $P = \vec{F} \cdot \vec{v}$.
7. What is the relation between kinetic energy $K$ and linear momentum $p$ for a mass $m$?
Explanation: Kinetic energy in terms of momentum is given by $K = \frac{1}{2}mv^2 = \frac{(mv)^2}{2m} = \frac{p^2}{2m}$.
8. Under what condition is total mechanical energy conserved?
Explanation: Total mechanical energy is conserved if only conservative forces perform work on the system.
9. The potential energy of a stretched spring with spring constant $k$ and elongation $x$ is:
Explanation: The elastic potential energy stored in a spring obeying Hooke's law is $U = \frac{1}{2}kx^2$.
10. What is the work done by the centripetal force on a particle moving with uniform speed in a circle?
Explanation: Since centripetal force is always perpendicular to the instantaneous displacement, $\theta = 90^\circ$ and $W = Fs \cos(90^\circ) = 0$.
11. Average power is defined as:
Explanation: Average power is the total work done divided by the total time interval: $P_{\text{avg}} = \frac{\Delta W}{\Delta t}$.
12. Mechanical efficiency $\eta$ is defined as the ratio of:
5. A block of mass $2\text{ kg}$ is dropped from a height of $5\text{ m}$ onto a spring of spring constant $k = 500\text{ N/m}$. The maximum compression of the spring is approximately:
Explanation: By conservation of mechanical energy, gravitational potential energy lost equals elastic potential energy gained: $mg(h + x) = \frac{1}{2}kx^2$. Solving for $x$ with $m=2, g=10, h=5, k=500$ gives $20(5 + x) = 250x^2 \Rightarrow 250x^2 - 20x - 100 = 0 \Rightarrow x \approx 0.5\text{ m}$.
6. Which of the following potential energy curves represents an unstable equilibrium point?
Explanation: An unstable equilibrium corresponds to a local maximum of potential energy where $\frac{dU}{dx} = 0$ and $\frac{d^2U}{dx^2} < 0$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. A body of mass $m$ accelerates uniformly from rest to velocity $v_1$ in time $t_1$. The instantaneous power delivered to the body as a function of time $t$ ($t \le t_1$) is:
Explanation: Acceleration $a = \frac{v_1}{t_1}$. Velocity at time $t$ is $v(t) = \frac{v_1}{t_1}t$. Force $F = ma = \frac{m v_1}{t_1}$. Power $P = F \cdot v(t) = \left(\frac{m v_1}{t_1}\right) \left(\frac{v_1}{t_1}t\right) = \frac{m v_1^2 t}{t_1^2}$.
2. A particle moves in a potential field $U(x) = \frac{a}{x^2} - \frac{b}{x}$. The equilibrium position $x_0$ is given by:
3. A bullet of mass $10\text{ g}$ moving with velocity $300\text{ m/s}$ hits a wooden plank and penetrates it to a depth of $5\text{ cm}$. The average resistance offered by the plank is:
Explanation: Work-Energy Theorem: $W_{\text{net}} = -F \cdot s = \Delta K = 0 - \frac{1}{2}mv^2$. Thus $F \cdot s = \frac{1}{2}(0.01)(300)^2 = 450\text{ J}$. $F = \frac{450}{0.05} = 9000\text{ N}$.
4. A force $\vec{F} = y\hat{i} + x\hat{j}$ acts on a particle. The work done in moving the particle from $(0,0)$ to $(1,1)$ along the path $y = x$ is:
Explanation: $W = \int (y dx + x dy)$. Along $y=x$, $dy = dx$. $W = \int_0^1 (x dx + x dx) = 2 \int_0^1 x dx = 2 \left[\frac{x^2}{2}\right]_0^1 = 1\text{ J}$.
5. An engine pumps water through a hose pipe. Water comes out with a velocity $v$ and mass per unit length of the jet is $\mu$. What is the power of the engine?
Explanation: Mass per second coming out $\frac{dm}{dt} = \mu v$. Kinetic energy per second (power) $P = \frac{1}{2} \left(\frac{dm}{dt}\right) v^2 = \frac{1}{2} (\mu v) v^2 = \frac{1}{2} \mu v^3$.
6. A block of mass $m$ slides down a rough inclined plane of inclination $\theta$ and length $L$ with constant velocity. The work done by friction is:
Explanation: Since velocity is constant, $\Delta K = 0$. By Work-Energy Theorem, $W_{\text{net}} = W_g + W_{\text{friction}} = 0$. $mgL \sin\theta + W_{\text{friction}} = 0 \implies W_{\text{friction}} = -mgL \sin\theta$.
7. The potential energy of a particle of mass $1\text{ kg}$ moving in a one-dimensional conservative field is given by $U(x) = \left(\frac{x^2}{2} - x\right)\text{ J}$. Its angular frequency of small oscillations about the stable equilibrium point is:
8. A force $F$ acting on a body depends on distance $x$ as $F = \frac{a}{\sqrt{x}}$. The work done in moving the body from $x = 0$ to $x = 4$ is proportional to:
Explanation: $W = \int_0^4 a x^{-1/2} dx = a \left[\frac{x^{1/2}}{1/2}\right]_0^4 = 2a \sqrt{4} = 4a$, which is proportional to $2$ (or $\sqrt{4}$). Note that $\sqrt{4} = 2$.
9. A ball is thrown vertically upwards with initial kinetic energy $E$. Due to air resistance, the mechanical energy decreases by $20\%$. What is its kinetic energy when it returns to the ground?
Explanation: Initial mechanical energy $E_i = E$ (taking potential energy at ground as 0). Final mechanical energy when it returns to ground (at $h=0$, $U=0$) is $E_f = K_f$. Since mechanical energy decreases by $20\%$, $E_f = E - 0.2E = 0.8E = K_f$.
10. If a machine has a mechanical efficiency of $75\%$ and delivers a useful output power of $3000\text{ W}$, what is the total input power?