The mechanics of astronomical objects to practical applications like wheels and turbines, rotational motion is a fundamental component. we will examine the dynamics of rotational motion about a fixed axis in this note.
Rotational Motion
A rigid body that rotates around a fixed axis is said to be in rotational motion. Around this axis, every particle in the body travels in a circle. Concepts like angular displacement, angular velocity, and angular acceleration which are analogous to linear motion control the dynamics of such motion.
The angle at which a point or line has been rotated in a particular direction around a given axis is known as the “angular displacement” (θ).
Measured in radians per second (rad/s), angular velocity (ω) is the rate at which angular displacement changes over time.
Measured in rad/s2, angular acceleration (α) is the rate at which angular velocity changes over time
Torque and Rotational Dynamics
In rotational motion, torque (τ) functions similarly to force in linear motion. It is a measurement of a force’s propensity to rotate a body about an axis. The force (F), distance from the axis of rotation (r), and angle (θ) between the force and the lever arm all affect torque: The direction of torque is determined using the right-hand rule
Moment of Inertia
The rotating equivalent of mass in linear motion is the moment of inertia (I). It measures a body’s resistance to angular acceleration about a certain axis. The mass distribution with respect to the axis of rotation determines the moment of inertia. The moment of inertia for a point mass is determined by:
For a rigid body, the total moment of inertia is the sum of contributions from all its particles: Some standard moments of inertia for common shapes include:
Solid sphere about its diameter:
Solid cylinder about its axis:
Thin rod about an axis perpendicular to its length through the center
Rotational Analog of Newton’s Second Law
The net torque operating on a body is equal to the product of its moment of inertia and angular acceleration, according to the rotational version of Newton’s second law: When it comes to solving rotational motion difficulties, this equation is essential
Work, Power, and Energy in Rotational Motion
Work Done by Torque: When a torque rotates a body through an angle, the work is done:
Rotational Kinetic Energy: A rotating body possesses kinetic energy given by: K = Mv2 / 2
Power in Rotational Motion: Power is the rate at which work is done:
Conservation of Angular Momentum
A body’s rotational motion is measured by its angular momentum (ℓ). Regarding a body that revolves around a fixed axis:
According to the conservation of angular momentum principle, a system’s angular momentum stays constant when no external torque occurs on it:
Applications for this idea are numerous and range from astrophysics to figure skating.
Rolling Motion
Translational and rotational motion combine to form rolling motion. When a body rolls without slipping, the following conditions must be met: where are the rolling object’s radius, angular velocity, and linear velocity of the center of mass?
Applications of Rotational Dynamics
1.Engineering: Design of gears, turbines, and flywheels relies heavily on the principles of rotational motion.
2. Sports: In games like discus throw or gymnastics, are torque and angular momentum is essential.
3. Astronomy: The rotational dynamics of planets and stars help explain phenomena like day-night cycles and stellar formation.
.
Key points to remember:-
In order to calculate the torque on a rigid body:-
Consider only those forces that lie in plane perpendicular to the axis, Forces which are parallel to the axis will give the torques perpendicular to the axis and need not to be taken into account.
Consider only those components of the position vectors which are perpendicular to the axis. Components of position vectors along the axis will the result in torques perpendicular to the axis and need not to be taken into account.
Work done by a torque :- Work done by a force F1 acting on a particle of a body rotating about a fixed axis; the particle describes a circular path with centre C on the axis; gives the displacement of the particle. dw1 = td theta
For more than one forces,
dw = (t1 +t2 + ..) d theta= td theta power p = dw / dt = t d theta / dt = t omega [i]
Rate of change of kinetic energy d/ dt [I omega2 / 2] =
(2 omega/ 2 d omega/dt) = 1 omega Alpha [ii]
From eqn [i] and [ii] t = I Alpha
Comparison of translational and rotational motion
Sl No | Linear Motion | Rotational motion about a Fixed Axis |
1 | Displacement x | Angular displacement theta |
2 | Velocity v= dx /dt | Angular velocity omega = |
3 | Acceleration a = dv/dt | Angular acceleration Alpha d omega/ dt |
4 | Mass M | Moment of inertia I |
5 | Force F = Ma | Torque t = I alpha |
6 | Work W = Fs | Work W = td theta |
7 | Power P = Fv | Power P = t omega |
8 | Linear momentum | Angular momentum I = I omega |
Note
The torque, moment of inertia, angular momentum, and their interactions in order to understand the dynamics of rotational motion about a fixed axis. These ideas serve as the foundation for examining and resolving rotational mechanics issues.
Angular displacement is the angle through which a point or line has been rotated about a fixed axis. Unlike linear displacement, which measures the straight-line distance between two points, angular displacement measures rotation in radians or degrees.
Moment of inertia is the measure of an object’s resistance to changes in its rotational motion. It plays a crucial role in determining how much torque is required to achieve a desired angular acceleration.
Torque is the rotational equivalent of force and is responsible for causing angular acceleration in a body. The greater the torque applied, the greater the change in angular motion.
Conservation of angular momentum states that if no external torque acts on a system, its angular momentum remains constant. This principle explains phenomena like a skater spinning faster when pulling their arms inward.
In rolling motion, the object rotates about its axis and translates linearly. The condition for rolling without slipping is that the linear velocity of the center of mass equals the product of the angular velocity and the radius.
Rotational dynamics is used in engineering designs (e.g., gears, turbines), sports (e.g., discus throw, gymnastics), and astronomy (e.g., planetary rotations and star formations).