Rotational motion is the motion of an object around a fixed axis of rotation. It is a type of motion that occurs when an object is rotating about a point, rather than moving in a straight line.
In rotational motion, an object is characterized by its angular position, angular velocity, and angular acceleration. The angular position of an object is its orientation with respect to a reference point and is measured in radians. Measured in radians per second, an object's angular velocity is the rate at which its angular position changes. The rate at which an object's angular velocity changes is called its angular acceleration, and it is expressed in radians per second squared.
Important Concept
An important concept in rotational motion is torque, which is a measure of the tendency of a force to rotate an object about an axis. It depends on the force's magnitude and the perpendicular distance from the axis of rotation to the line of action.
Where:
- τ = Torque
- F = Applied force
- r⊥ = Perpendicular distance from the axis of rotation to the line of action of the force
Another important concept in rotational motion is the moment of inertia. It is a measure of an object’s resistance to rotational motion. It depends on the distribution of mass and the distance of each particle from the axis of rotation. Objects with mass farther from the axis have greater inertia and resist changes in rotational motion more.
Rotational Motion: It is the motion of an object around a fixed axis. Its kinetic energy is
Angular Displacement
It is the measure of the change in an object’s orientation as it rotates about a fixed axis. Measured in radians, it is a vector quantity that indicates both the magnitude of rotation and its direction.
Angular Velocity
Angular velocity is the rate at which an object rotates, expressed in radians per second. It is a vector quantity describing the amplitude and direction of rotational motion.
The image below represents the tangential velocity and angular velocity (ω).
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Angular Acceleration
Angular acceleration is the rate at which angular velocity changes, expressed in radians per second squared. It is a scalar quantity that describes the rate at which rotational velocity changes.
Torque is a force that induces rotation and is equal to the product of the applied force and the perpendicular distance from the axis of rotation to the force's line of action. Torque can be used to explain an object's rotational motion since the net torque exerted on it is proportional to its angular acceleration.
Moment of Inertia
An object's moment of inertia is a measure of its resistance to rotational motion that depends on its mass distribution and the distance of each particle from the axis of rotation. The moment of inertia can be used to characterize an object's rotational motion since the net torque acting on an object is proportional to its angular acceleration, which is inversely proportional to the moment of inertia.
Kinetic Energy of Rotation
The kinetic energy of rotation is the energy that an object has as a result of its rotational motion, which is equal to one-half the product of its moment of inertia and angular velocity squared. This energy can be used to describe an object's rotational motion, as an object with a high angular velocity and a large moment of inertia will have a high kinetic energy of rotation.
Examples,
- Earth rotates around its axis.
- A spinning top.
- An axle rotates a wheel.
- A figure skater is spinning on the ice.
Formulas used in Rotational Motion
The study of particle systems and rotational motion employs several formulas. Some of the most important formulas are,
Center of Mass
The formula for the centre of mass of a system of particles is given by,
\mathbf{R}_{\text{cm}} = \frac{m_1 \mathbf{r}_1 + m_2 \mathbf{r}_2 + \dots + m_n \mathbf{r}_n}{m_1 + m_2 + \dots + m_n} where,
Rcm is the position vector of the centre of mass,
m1, m2, ..., mn are the masses of the particles,
r1, r2, ..., rn are the position vectors of the particles.
Linear Momentum
The formula for the linear momentum of a system of particles is given.
\mathbf{P} = m_1 \mathbf{v}_1 + m_2 \mathbf{v}_2 + \dots + m_n \mathbf{v}_n where,
P is the linear momentum of the system,
m1, m2, ..., mn are the masses of the particles,
v1, v2, ..., vn are the velocity vectors of the particles.
Linear Kinetic Energy
The formula for the linear kinetic energy of a system of particles is given.
T = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 + \dots + \frac{1}{2} m_n v_n^2 where
T is the linear kinetic energy of the system,
m1, m2, ..., mn are the masses of the particles,
v1, v2, ..., vn are the velocity vectors of the particles.
Torque
The formula for the torque on a particle is given.
\tau = r \times F where
τ is the torque,
r is the position vector of the particle with respect to the axis of rotation,
F is the force applied to the particle.
Angular Momentum
The formula for the angular momentum of a system of particles is given.
L = I \, \omega where,
L is the angular momentum of the system,
I is the moment of inertia of the system,
ω is the angular velocity of the system.
Angular Kinetic Energy
The formula for the angular kinetic energy of a system of particles is given.
K = \frac{1}{2} I \omega^2 where
K is the angular kinetic energy of the system,
I is the moment of inertia of the system,
ω is the angular velocity of the system.
These formulas provide a foundation for the study of systems of particles and rotational motion and are used to analyze the motion of rotating objects and to predict the behavior of rotating systems.
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Solved Problems
Question 1: A system of three particles with masses m1 = 3 kg, m2 = 4 kg, and m3 = 5 kg is located at positions r1 = (2, 3, 0), r2 = (4, 0, 0), and r3 = (0, 4, 0), respectively. Find the position of the center of mass.
Solution: Position of the centre of mass can be found using the formula,
Rcm =
\frac{m_1 r_1 + m_2 r_2 + m_3 r_3}{m_1 + m_2 + m_3} Rcm
= \frac{3(2,3,0) + 4(4,0,0) + 5(0,4,0)}{3+4+5} Rcm
= \frac{(6,9,0) + (16,0,0) + (0,20,0)}{12} Rcm
= \frac{(22,29,0)}{12}
= \left(\frac{11}{6}, \frac{29}{12}, 0\right) So the centre of mass of the system is located at
= \left(\frac{11}{6}, \frac{29}{12}, 0\right)
Question 2: A system of two particles with masses m1 = 2 kg and m2 = 3 kg is moving with velocities v1 = (4, 5, 0) m/s and v2 = (3, 2, 0) m/s, respectively. Find the linear momentum of the system.
Solution: Linear momentum of the system can be found using the formula,
P = m1v1 + m2v2
P = 2(4, 5, 0) + 3(3, 2, 0)
P = (8, 10, 0) + (9, 6, 0)
P = (17, 16, 0) kgms-1
So, linear momentum of the system is (17, 16, 0) kgms-1
Question 3: A system of two particles with masses m1 = 2 kg and m2 = 3 kg is rotating with angular velocities ω1 = 2 rad/s and ω2 = 3 rad/s, respectively. The moments of inertia of the particles are I₁ = 5 kg. · m² and I2 = 7 kg · m². Find the angular momentum of the system.
Solution: Angular Momentum of the system can be found using the formula:
L = I_1 \omega_1 + I_2 \omega_2 Substitute the values
L = 5 \times 2 + 7 \times 3
L = 10 + 21 L = 31 kg m2/s
So the Angular Momentum of the system is 31 kg m2/s
Question 4: A system of two particles with masses m1 = 2 kg and m2 = 3 kg is rotating with angular velocities ω1 = 2 rad/s and ω2 = 3 rad/s, respectively. The moments of inertia of the particles are I₁ = 5 kg. · m² and I2 = 7 kg · m². Find the angular kinetic energy of the system.
Solution: Angular Kinetic Energy of the system can be found using the formula,
K = \frac{1}{2} I_1 \omega_1^2 + \frac{1}{2} I_2 \omega_2^2 Substitute the values
K = \frac{1}{2} \times 5 \times 2^2 + \frac{1}{2} \times 7 \times 3^2
K = \frac{1}{2} \times 20 + \frac{1}{2} \times 63
K = 10 + 31.5 So the Angular Kinetic Energy of the system is
K = 41.5\ \text{J}
Unsolved Problems
Question 1: A wheel of mass 10 kg and radius 0.5 m is rotating about its axis with angular velocity 4 rad/s. Calculate its rotational kinetic energy.
Question 2: Two particles, m1 = 3 kg and m2 = 2 kg, are attached to the ends of a light rod of length 2 m. The rod rotates about its center. Find the moment of the system's inertia.
Question 3: A torque of 15 N·m acts on a disc of moment of inertia 5 kg·m². Find its angular acceleration.
Question 4: A system of three particles with masses of 2 kg, 3 kg, and 4 kg is at distances of 1 m, 2 m, and 3 m from a fixed axis of rotation. The system turns at a rate of 5 rad/s. Find the total angular momentum of the system.
Question 5: A solid cylinder of mass 12 kg and radius 0.3 m is rolling without slipping on a horizontal surface with linear speed 6 m/s. Find its total kinetic energy.