Rotational Kinematics Formula
The rotation which is around a fixed axis is a special case of motion which is known as the rotational motion. According to the rotation of Euler's theorem, we can say that the simultaneous rotation which is along with a number of stationary axes at the same time is impossible. But only if two rotations are forced at the same time, a new axis of rotation will appear to us. Here we assume that the rotation is also stable such that no torque is required to keep it going on and on.
Kinematics of Rotational Motion
The rotation or we can say that the kinematics and dynamics that is of rotation around a fixed axis of a rigid body are mathematically much simpler than those for free rotation of a rigid body. They are said to be entirely analogous to those of linear motion along a single or a fixed direction which is not true for the free rotation that too of a rigid body. The expressions which are given for the kinetic energy of the object and we can say for the forces on the parts of the object are also said to be simpler for rotation around a fixed axis. We can say that which is closer than for general rotational motion.
For these reasons we can say that the rotation around a fixed axis is typically taught in introductory physics courses that are after students have mastered linear motion. The full generality is that rotational motion is not usually taught in introductory physics classes.
A body which is rigid is an object of finite extent in which all the distances in between the component particles are constant. No truly rigid body it is said to exist amid external forces that can deform any solid. For our purposes as we know that then a rigid body which is a solid which requires large forces to deform it appreciably.
A change that we have seen in the position of a particle in three-dimensional space that can be completely specified by three coordinates. A change that is in the position of a body which is rigid is more is said to be complicated to describe. It can be said that it is regarded as a combination of two distinct types of motion which is translational motion and circular motion.
Purely which is said to be a translational motion generally occurs when every particle of the body has the same amount of instantaneous velocity as every other particle. We can say that, the path which is traced out by any particle that is exactly said to be parallel to the path which is traced out by every other particle in the body.
Axis of Rotation
We can say that the rotational motion occurs if every particle in the body moves in a circle about a single line. This line is known as the axis of rotation. Then the radius which is vectors from the axis to all particles which undergo the same angular displacement at the same time. The axis of rotation need not go through the body. In general, we can say that any rotation can be specified completely by the three angular displacements we can say that with respect to the rectangular-coordinate axes x, y, and z. Any change that is in the position which is of the rigid body. Thus, we can say that this is described by three translational and three rotational coordinates.
Any displacement which is of a body that is rigid may be arrived at by first subjecting the body to a displacement that is followed by a rotation or we can say is conversely to a rotation which is followed by a displacement. We already know that for any collection of particles whether it is at rest with respect to one another as in a rigid body or we can say in relative motion like the exploding fragments that is of a shell and then the acceleration which is of the centre of mass is given by the following equation:
\[F_{net} = Ma_{cm}\]
where capital letter M is the total mass of the system and acm is said to be the acceleration which is of the centre of mass.
Rotational Motion
The above development that we have known is a special case of general rotational motion. In the general case, we can say that angular displacement and angular velocity, angular acceleration and torque are considered to be vectors.
An angular displacement which we already know is considered to be a vector which is pointing along the axis that is of magnitude equal to that of A right-hand rule which is said to be used to find which way it points along the axis we know that if the fingers of the right hand are curled to point in the way that the object has rotated and then the thumb which is of the right-hand points in the direction of the vector.
If we take a disk that spins counterclockwise as seen from above it is said to be the angular velocity vector that points upwards.
FAQs on Kinematics Rotational Motion Around Fixed Axis
1. What is meant by the kinematics of rotational motion about a fixed axis?
Kinematics of rotational motion describes the motion of a rigid body rotating around a stationary axis. It focuses purely on the geometry of motion—angular displacement (θ), angular velocity (ω), and angular acceleration (α)—without considering the forces or torques that cause the rotation. In this type of motion, every particle of the body moves in a circle, and the centre of each circle lies on the fixed axis of rotation.
2. What are the three fundamental equations for rotational kinematics under constant angular acceleration?
For a rigid body rotating about a fixed axis with a constant angular acceleration (α), the motion can be described by three key equations, which are analogous to linear motion equations:
- ω = ω₀ + αt: This equation relates the final angular velocity (ω) to the initial angular velocity (ω₀), angular acceleration (α), and time (t).
- θ = θ₀ + ω₀t + ½αt²: This equation calculates the final angular position (θ) based on the initial position (θ₀), initial angular velocity, acceleration, and time.
- ω² = ω₀² + 2α(θ - θ₀): This equation connects final angular velocity with initial angular velocity, acceleration, and the total angular displacement (θ - θ₀).
3. What are some common real-world examples of rotation around a fixed axis?
Rotation around a fixed axis is observed in many everyday objects. Common examples include:
- The blades of a ceiling fan rotating around its central motor.
- A door swinging on its hinges, where the line of the hinges acts as the fixed axis.
- A spinning top rotating rapidly about its vertical axis (before it starts wobbling).
- The hands of an analogue clock moving around the centre point.
- A Ferris wheel turning around its central axle.
4. How do angular velocity and linear velocity differ for various particles in a rigid body rotating about a fixed axis?
For a rigid body rotating around a fixed axis, all particles have the same angular velocity (ω) and same angular acceleration (α) because the body rotates as a single unit. However, their linear velocity (v) is different. The linear velocity of any particle is given by the formula v = rω, where 'r' is the perpendicular distance of the particle from the axis of rotation. This means particles farther from the axis travel a larger circular path in the same amount of time and thus have a higher linear velocity than particles closer to the axis.
5. What is the relationship between torque, moment of inertia, and angular acceleration?
The relationship is described by the rotational analogue of Newton's second law: τ = Iα. Here, torque (τ) is the rotational equivalent of force, which causes a change in rotational motion. The moment of inertia (I) is the rotational equivalent of mass, representing an object's resistance to angular acceleration. The angular acceleration (α) is the rate of change of angular velocity. This equation shows that the net torque on an object is directly proportional to the angular acceleration it experiences.
6. Why are the kinematic equations for fixed-axis rotation directly analogous to those for linear motion?
The remarkable similarity exists because the fundamental concepts are parallel. In linear motion, we track displacement (s), velocity (v), and acceleration (a). In fixed-axis rotation, all particles share a common angular displacement (θ), angular velocity (ω), and angular acceleration (α). This allows for a direct substitution of variables:
- Linear displacement (s) corresponds to angular displacement (θ).
- Linear velocity (v) corresponds to angular velocity (ω).
- Linear acceleration (a) corresponds to angular acceleration (α).
- Mass (m), or inertia, corresponds to moment of inertia (I).
Because the underlying mathematical relationships for rates of change are identical, the structure of the equations remains the same.
7. How does the distribution of an object's mass affect its rotation?
The distribution of mass is crucial and is quantified by the moment of inertia (I). For a given mass, if the mass is concentrated far from the axis of rotation, the moment of inertia is large. This makes the object harder to start rotating (requires more torque for the same angular acceleration) and harder to stop once it is rotating. Conversely, if the same mass is concentrated close to the axis of rotation, the moment of inertia is small, and it is much easier to change its state of rotation. This is why a figure skater pulls their arms in to spin faster.
8. What is the difference between an ideal 'fixed axis' in physics and the 'instantaneous centre of rotation' in biomechanics?
An ideal 'fixed axis' in physics is a perfectly stationary, one-dimensional line around which a rigid body rotates. Every point on the body maintains a constant distance from this axis. In contrast, the joints in a human body, like the knee or shoulder, do not have a truly fixed axis. Due to the complex interaction of bones, ligaments, and muscles, the centre of rotation shifts slightly during movement. This moving axis is referred to as an 'instantaneous centre of rotation' (ICR), representing the point around which the body segment appears to be rotating at a specific moment in time.
9. How are the rotational kinematic equations derived using calculus?
The rotational kinematic equations can be derived from the fundamental definitions of angular velocity and angular acceleration using integration, assuming constant acceleration.
- We start with the definition of angular acceleration: α = dω/dt. Rearranging and integrating gives ∫dω = ∫α dt, which results in ω = ω₀ + αt.
- Next, we use the definition of angular velocity: ω = dθ/dt. Substituting the first equation, we get dθ = (ω₀ + αt)dt. Integrating this expression gives θ = θ₀ + ω₀t + ½αt².
- Finally, by using the chain rule (α = dω/dt = (dω/dθ)(dθ/dt) = ω(dω/dθ)), we can write α dθ = ω dω. Integrating this yields the third equation: ω² = ω₀² + 2α(θ - θ₀).
