Home › Physics › Mechanics › Torque and Rotational Dynamics
Mechanics
Torque and Rotational Dynamics
Torque is the rotational analogue of force, defined as the cross product of the position vector and the applied force about a given point or axis. It measures the effectiveness of a force in producing rotational acceleration about an axis of rotation.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of torque, relation with angular acceleration, principle of moments, conditions for static equilibrium of rigid bodies.
PGT extension
Vector formulation, torque in polar coordinates, central forces, torque on a current loop, torque about a moving point, and Lagrangian formulation.
Prerequisites
Vector cross product properties
Newton's laws of motion
Centre of mass and moment of inertia
Definition and Physical Significance
Torque (tau) is a vector quantity that represents the turning effect of a force.
It depends on the magnitude of the force, the moment arm (perpendicular distance from the axis of rotation to the line of action of the force), and the angle between them.
The SI unit of torque is Newton-metre (N·m) and its dimensional formula is [M L2 T-2].
Torque and Angular Acceleration
For a rigid body rotating about a fixed axis, the net external torque is equal to the product of the moment of inertia (I) and the angular acceleration (alpha).
Expressed as tau = I alpha, this relation is the rotational counterpart of Newton's second law (F = ma).
If the net external torque acting on a system is zero, the angular acceleration is zero, leading to a constant angular velocity.
Conditions for Static Equilibrium
For a rigid body to be in complete mechanical equilibrium, both the vector sum of all external forces and the vector sum of all external torques about any point must vanish.
Sum of forces: Sigma F = 0 ensures translational equilibrium.
Sum of torques: Sigma tau = 0 ensures rotational equilibrium.
Torque due to Central Forces
A central force is directed toward or away from a fixed center point, and its line of action passes through that center.
Since the position vector and the central force are collinear (angle is 0 or pi radians), the torque exerted by a central force about the center of force is identically zero.
Zero torque implies the conservation of angular momentum for any particle moving under a central force field.
Derivations
Relation between Torque and Angular Momentum
Conditions: Particle or system of particles with respect to a fixed origin
Start with the definition of angular momentum L = r x p = r x (mv).
Take the time derivative of angular momentum: dL/d$t = d/dt (r x p) = (dr/dt x p) + (r x dp/dt)$.
Recognize that dr/dt = v and p = mv, so dr/dt x p = v x (mv) = 0 because a vector cross product with itself is zero.
Substitute dp/d$t = F (Newton's second law) into the second term: r x (dp/dt) = r x F = \tau$.
Conclude that the time rate of change of angular momentum equals the net external torque acting on the system: dL/dt = tau.
Result: dL/dt = tau
Work Done and Power in Rotational Motion
Conditions: Rigid body rotating about a fixed axis under constant or variable torque
Express the infinitesimal work done by a torque as dW = tau d(theta), where tau is torque and d(theta) is angular displacement.
Integrate the work from an initial angle theta1 to a final angle theta2: $W = integral from \theta_{1}$ to theta2 of (tau d(theta)).
To find rotational power (P), differentiate work with respect to time: $P = dW/dt = \tau (d(\theta)/dt)$.
Substitute angular velocity omeg$a = d(\theta)/dt into the power equation$.
Result: P = tau \omega
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Vector Torque
$$\vec{\tau} = \vec{r} \times \vec{F}$$
r = position vector, F = applied force N·m
General definition
Scalar Magnitude of Torque
$$\tau = r F \sin(\theta)$$
r = distance from axis, F = force magnitude, theta = angle between r and F N·m
Planar geometry
Torque and Moment of Inertia
$$\tau = I \alpha$$
I = moment of inertia, alpha = angular acceleration N·m
Fixed axis rotation
Rate of Change of Angular Momentum
$$\vec{\tau} = \frac{d\vec{L}}{dt}$$
L = angular momentum, t = time N·m (or kg·m2s-2)
Inertial frame of reference
Rotational Work Done
$$W = \int \tau d\theta$$
theta = angular displacement Joule (J)
Rigid body rotation
Rotational Power
$$P = \tau \omega$$
omega = angular velocity Watt (W)
Constant or instantaneous torque
Principle of Moments (Equilibrium)
$$F_1 r_1 = F_2 r_2$$
F = forces, r = moment arms on opposite sides of pivot N·m
Static equilibrium on a lever
Torque in Polar Coordinates
$$\tau_z = r F_\theta$$
$F_theta = transverse component of force$ N·m
2D polar coordinate system
Graphs and experiments
Torque vs. Angular Acceleration Graph
Draw: A straight line passing through the origin with Torque (tau) on the vertical axis and Angular Acceleration (alpha) on the horizontal axis.
Interpret: The slope of this linear graph directly yields the Moment of Inertia (I) of the rotating body.
Exam use: Tested conceptually to find moment of inertia from experimental data or theoretical curves.
Searle's or Flywheel Apparatus Experiment
Draw: A schematic diagram of a heavy flywheel mounted on horizontal frictionless bearings with a cord wound around its axle carrying a hanging mass.
Interpret: Used to determine the moment of inertia of the flywheel by applying a known gravitational torque (mgr) and measuring angular acceleration.
Exam use: Questions focus on energy conservation, frictional torque correction, and calculation of moment of inertia.
Torque as a Function of Angle (Work Done Graph)
Draw: A curve plotting Torque (tau) versus Angular Displacement (theta).
Interpret: The area under the torque-angular displacement curve represents the total work done in rotating the body.
A uniform rod of mass M = 3 kg and length L = 2 m is pivoted smoothly at one end. It is released from rest in the horizontal position. Find the initial angular acceleration of the rod.
The weight of the rod acts downwards at its centre of mass, located at a distance $r = L/2 from the pivot$.
The magnitude of the gravitational torque about the pivot is ta$u = (Mg) * (L/2)$.
The moment of inertia of a uniform rod about an axis passing through one end is $I = (1/3) M L^2$.
Using tau = I alpha, equate the expressions: (Mg L)/$2 = ((1/3) M L^2) \alpha$.
A particle of mass 2 kg moves in the xy-plane with a position vector $r = (3t i + 4t^2 j) m$. Find the torque acting on the particle about the origin at t = 1 s.
Find the velocity vector by differentiating position with respect to time: $v = dr/dt = (3 i + 8t j) m/s$.
At t = 1 s, velocity $v = (3 i + 8 j) m/s$.
Calculate linear momentum $p = m v = 2 * (3 i + 8 j) = (6 i + 16 j) kg·m/s$.
Find position vector at $t = 1 s: r = (3(1) i + 4(1)^2 j) = (3 i + 4 j) m$.
Calculate torque as the rate of change of angular momentum or evaluate force if acceleration is known. Alternatively, find acceleration $a = dv/dt = 8 j m/s^2$, so force $F = m a = 2 * (8 j) = 16 j N$.
Calculate torque ta$u = r x F = (3 i + 4 j) x (16 j)$.
Compute cross product: (3 * 16)(i x j) + (4 * 16)(j x j) = 48 k N·m.
Answer: 48 k N·m
Assumptions and limitations
Rigid body assumption: distances between constituent particles do not change under applied torque, which neglects internal deformations.
Common misconceptions
Common error: Torque is a property of a single point in space.
Correct: Torque is always defined with respect to a specific reference point or axis of rotation.
The position vector r depends entirely on the choice of the origin/pivot.
Common error: If the net force on a body is zero, the body cannot rotate.
Correct: A body can experience zero net force yet have a non-zero net torque (forming a couple), resulting in angular acceleration.
Translational equilibrium (Sigma F = 0) does not guarantee rotational equilibrium (Sigma tau = 0).
Quick revision
Torque is the rotational analogue of force, given by tau = r x F = I alpha.
Scalar Magnitude of Torque: \tau = r F \sin(\theta)
Must remember
Torque $\vec{\tau}$ is a vector quantity defined as the cross product of position vector and force: $\vec{\tau} = \vec{r} \times \vec{F}$.
The SI unit of torque is Newton-metre ($\text{N}\cdot\text{m}$) and its dimensional formula is $[M L^2 T^{-2}]$.
The rotational counterpart of Newton's second law for a fixed axis is $\tau = I\alpha$, where $I$ is the moment of inertia and $\alpha$ is the angular acceleration.
For complete mechanical equilibrium of a rigid body, both $\sum \vec{F} = 0$ and $\sum \vec{\tau} = 0$ must be satisfied.
Central forces exert zero torque about the center of force because the position vector and the force are collinear, leading to the conservation of angular momentum.
Glossary
TorqueA vector quantity representing the turning effect of a force, given by $\tau = r F \sin(\theta)$.
Moment ArmThe perpendicular distance from the axis of rotation to the line of action of the applied force.
Moment of InertiaA quantity expressing a body's tendency to resist angular acceleration, denoted by $I$.
Static EquilibriumA state in which a rigid body is at rest with both net external force and net external torque equal to zero.
Central ForceA force whose line of action always passes through a fixed center point.
Translational vs Rotational Dynamics
Parameter
Translational Motion
Rotational Motion
Displacement
Linear displacement ($x$)
Angular displacement ($\theta$)
Inertia
Mass ($m$)
Moment of Inertia ($I$)
Force / Torque
Force ($F = ma$)
Torque ($\tau = I\alpha$)
Momentum
Linear momentum ($p = mv$)
Angular momentum ($L = I\omega$)
Practice Quiz
Choose an option to reveal the explanation.
1. What is the dimensional formula for torque?
Explanation: Torque is force multiplied by distance: $[M L T^{-2}] \times [L] = [M L^2 T^{-2}]$.
2. Which of the following conditions is necessary and sufficient for a rigid body to be in complete mechanical equilibrium?
Explanation: Complete mechanical equilibrium requires both translational equilibrium ($\sum \vec{F} = 0$) and rotational equilibrium ($\sum \vec{\tau} = 0$).
3. What is the torque exerted by a central force about the center of force?
Explanation: A central force is collinear with the position vector, making the angle $\theta = 0$ or $\pi$, so $\tau = rF\sin(0) = 0$.
4. If the net external torque acting on a system is zero, which of the following quantities remains conserved?
Explanation: Since $\vec{\tau} = \frac{d\vec{L}}{dt}$, if the net external torque is zero, the rate of change of angular momentum is zero, making angular momentum conserved.
5. What is the SI unit of torque?
Explanation: Torque is the product of force (Newtons) and distance (metres), giving the SI unit $\text{N}\cdot\text{m}$.
6. In planar geometry, if a force $F$ is applied at an angle $\theta$ to the position vector $r$, the scalar magnitude of torque is given by:
Explanation: The magnitude of the cross product is given by $\tau = r F \sin(\theta)$.
7. For a fixed-axis rotation, the relation between torque and angular acceleration is expressed as:
Explanation: The rotational analogue of $F = ma$ for a rigid body is $\tau = I\alpha$.
8. A body experiences zero net force, but a non-zero net torque. What will happen to the body?
Explanation: Zero net force means no linear acceleration ($\sum \vec{F} = 0$), while non-zero net torque causes angular acceleration ($\sum \vec{\tau} \neq 0$).
9. What is the rotational work done by a torque $\tau$ for an angular displacement $\theta$?
Explanation: Rotational work is defined as the integral of torque with respect to angular displacement: $W = \int \tau d\theta$.
10. Rotational power $P$ in terms of torque $\tau$ and angular velocity $\omega$ is given by:
Explanation: Rotational power is the product of torque and angular velocity: $P = \tau \omega$.
11. According to the principle of moments for a lever in static equilibrium, if $F_1$ and $F_2$ are forces at distances $r_1$ and $r_2$ from the pivot on opposite sides, then:
Explanation: For rotational equilibrium, clockwise torque equals counter-clockwise torque, so $F_1 r_1 = F_2 r_2$.
12. In a 2D polar coordinate system, the z-component of torque is expressed as:
Explanation: In 2D polar coordinates, torque about the origin is contributed by the transverse component of force: $\tau_z = r F_\theta$.
13. Why is torque always defined with respect to a specific reference point or axis of rotation?
Explanation: The position vector $\vec{r}$ in $\vec{\tau} = \vec{r} \times \vec{F}$ is measured relative to a chosen reference point/origin.
14. What is the relation between torque and angular momentum in an inertial frame of reference?
Explanation: Newton's second law for rotation states that the net external torque equals the rate of change of angular momentum: $\vec{\tau} = \frac{d\vec{L}}{dt}$.
15. If the net external torque on a rigid body is zero, what can be said about its angular velocity?
Explanation: Zero net torque implies zero angular acceleration ($\alpha = 0$), which means angular velocity remains constant.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A force $\vec{F} = (2\hat{i} + 3\hat{j})\text{ N}$ is applied at a position vector $\vec{r} = (1\hat{i} - 2\hat{j})\text{ m}$. What is the torque acting on the particle?
2. A uniform ladder of length $L$ and weight $W$ rests against a smooth vertical wall with its lower end on a rough horizontal floor. For static equilibrium, the frictional force at the floor must be equal in magnitude to:
Explanation: For horizontal equilibrium, the frictional force at the floor must balance the normal reaction force exerted by the smooth vertical wall.
3. A particle moves in a central force field. Which of the following is true regarding its motion?
Explanation: Since a central force exerts zero torque about the center of force, angular momentum remains conserved.
4. A wheel of moment of inertia $2\text{ kg}\cdot\text{m}^2$ is rotating at an angular velocity of $10\text{ rad/s}$. A constant retarding torque of $4\text{ N}\cdot\text{m}$ is applied to stop it. What is the time taken to bring the wheel to rest?
5. A couple consists of two equal and opposite forces separated by a perpendicular distance $d$. The magnitude of the torque of the couple is independent of:
Explanation: The torque of a couple is independent of the choice of origin because the sum of the forces is zero.
6. If a rigid body is acted upon by three non-parallel forces in static equilibrium, then the lines of action of these three forces must be:
Explanation: For three coplanar forces to maintain static equilibrium, they must either be parallel or meet at a single point (concurrent).
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. What is the angle between the position vector $\vec{r}$ and force $\vec{F}$ for maximum torque magnitude?
Explanation: Since $\tau = rF\sin(\theta)$, torque is maximum when $\sin(\theta) = 1$, which occurs at $\theta = 90^\circ$.
2. A constant torque of $50\text{ N}\cdot\text{m}$ turns a wheel about its fixed axis through an angle of $2\text{ radians}$. What is the work done?
3. Which physical law is represented by the equation $\vec{\tau} = \frac{d\vec{L}}{dt}$?
Explanation: This is the general vector form of Newton's second law for rotational motion in an inertial frame.
4. Two children of weights $300\text{ N}$ and $400\text{ N}$ sit on opposite sides of a seesaw. If the $300\text{ N}$ child is at a distance of $2\text{ m}$ from the pivot, where must the $400\text{ N}$ child sit for rotational equilibrium?
Explanation: Using the principle of moments: $F_1 r_1 = F_2 r_2 \Rightarrow 300 \times 2 = 400 \times r_2 \Rightarrow r_2 = 600 / 400 = 1.5\text{ m}$.
5. In 2D polar coordinates, if a force has only a radial component ($F_\theta = 0$), what is the torque about the origin?
Explanation: Since $\tau_z = r F_\theta$, if $F_\theta = 0$, the torque is identically zero.
6. A rigid body is in translational equilibrium if:
Explanation: Translational equilibrium is defined strictly by $\sum \vec{F} = 0$.
7. What happens to the angular acceleration of a rigid body if the net external torque is doubled while keeping the moment of inertia constant?
Explanation: From $\alpha = \tau / I$, angular acceleration is directly proportional to the net external torque.
8. Which of the following is NOT a condition for static equilibrium of a rigid body?
Explanation: Static equilibrium requires angular velocity to be zero (or constant, but specifically zero for static equilibrium), not necessarily maximum.
9. The moment of inertia of a body depends upon:
Explanation: Moment of inertia depends on the mass and how that mass is distributed relative to the axis of rotation.
10. A torque of $20\text{ N}\cdot\text{m}$ produces an angular acceleration of $4\text{ rad/s}^2$ in a rigid body. What is the moment of inertia of the body?