Rigid Body :-
In physics, a rigid body is an idealized solid object in which deformation (change in shape or size) is negligible. This means that the distance between any two given points on the body remains constant over time, regardless of external forces or motion.
Characteristics of a Rigid Body:
- No Deformation: The relative positions of all particles within the body remain fixed.
- Mass Distribution: The body has a defined shape, size, and mass distribution.
- Motion Types: A rigid body can undergo two types of motion:
Mathematical Definition:
For any two points ( P ) and ( Q ) in a rigid body, the distance between them remains unchanged:
For any two points \( P \) and \( Q \) in a rigid body, the distance between them remains unchanged:
\[ | \vec{r}_P – \vec{r}_Q | = \text{constant} \]
where \( \vec{r}_P \) and \( \vec{r}_Q \) are the position vectors of the points.
The Rotational Vairiable
Axis of rotaion
The axis of rotation is the straight line through a rigid body (or through space fixed relative to the body) that remains stationary while all other points of the body move in circular paths around it.
Examples
- Earth’s Rotation: The Earth is a rigid body rotating around an axis that passes through the North and South Poles. This is a nearly fixed axis.
- Spinning Top: The axis of a spinning top is the sharp point at the bottom and the centerline through its body. This axis itself often rotates (a motion called precession), demonstrating an instantaneous axis that changes.
- A Rolling Bicycle Wheel: The instantaneous axis of rotation is the line where the tire touches the road. At any given moment, every point on the wheel is rotating around that line of contact.
- A Door: The axis of rotation is the straight line through the hinges.
Angular displacement
Angular displacement is defined as “the angle in radians ( or degrees or revolutions) through which a point or line has been rotated in a specified sense about a specified axis of Rotation. It is the angle of the movement of a body in a circular path.
For example, in the diagram below, a rigid body rotates about an axis in an anticlockwise direction. A point P on the rigid body moves to a new position Q. The angle swept by the line connecting point P to the axis is Δθ (delta theta), which is known as the Angular Displacement.
Average Angular velocity
Average Angular Velocity of a rotating rigid body is the ratio of the angular displacement to the time interval in which displacement occures.
Mathematically, it is defined as the ratio of the angular displacement (Δθ) to the time interval (Δt) taken for that displacement.
The formula is:
\[ \omega_{\text{avg}} = \frac{\Delta \theta}{\Delta t} \]Where
\(\omega_{avg}\) = average angular velocity (SI unit is radians per second, rad/s)
\(\Delta \theta\) = change in angular position (angular displacement in radians, rad)
\(\Delta t\) = change in time (in seconds, s)
Instantaneous Angular velocity
The instantaneous angular velocity of a rotating body is the rate of change of angular displacement with respect to time, at a particular instant.
Mathematically, it is defined as:
\[ \vec{\omega}_{\text{inst}} = \lim_{\Delta t \to 0} \frac{\overrightarrow{\Delta \theta}}{\Delta t} = \frac{\overrightarrow{d\theta}}{dt} \]
where
\(\vec{\omega}_{\text{inst}}\) = average angular velocity (SI unit is radians per second, rad/s)
\(\overrightarrow{d\theta}\) = small change in angular position (SI unit is radians, rad)
\(dt\) = small change in time (in seconds, s)
- Direction: The direction of the angular velocity( instantaneous Angular Velocity ) is given by the right-hand rule: if you curl the fingers of your right hand in the direction of rotation, your thumb points in the direction of the angular velocity.
- Sense: In your diagram, when the rotation is anticlockwise, the vector \(\vec{\omega}_{\text{inst}}\) points out of the screen/page toward you. Conversely, if the rotation is clockwise, the vector \(\vec{\omega}_{\text{inst}}\) points into the screen/page away from you.
- SI unit of Instantaneous Angular Velocity is rad/sec.
Rotation with Constant Angular Acceleration
The equations for rotational motion with constant angular acceleration describe the relationships between angular displacement, velocity, and acceleration. These are analogous to the linear kinematic equations.
Angular velocity-time equation:
\[ \omega = \omega_0 + \alpha t \]where ω is final angular velocity, ω0 is initial angular velocity, \(\alpha \) is constant angular acceleration, and t is time.
\[ \alpha = \frac{d\omega}{dt} \] \[ d\omega = \alpha \, dt \] \[ \int_{\omega_0}^{\omega} d\omega = \alpha \int_{0}^{t} dt \] \[ \omega – \omega_0 = \alpha t \] \[ \omega = \omega_0 + \alpha t \] — \[ d\theta = \omega \, dt \] \[ d\theta = \left( \omega_0 + \alpha t \right) dt \] \[ \int d\theta = \omega_0 \int_0^t dt + \alpha \int_0^t t \, dt \] \[ \theta – 0 = \omega_0 t + \tfrac{1}{2}\alpha t^2 \] \[ \theta = \omega_0 t + \tfrac{1}{2}\alpha t^2 \] — \[ \alpha = \frac{d\omega}{dt} \] \[ \alpha = \frac{d\omega}{d\theta} \cdot \frac{d\theta}{dt} \] \[ \alpha = \omega \cdot \frac{d\omega}{d\theta} \] \[ \int \omega \, d\omega = \alpha \int d\theta \] \[ \int_{\omega_0}^{\omega} \omega \, d\omega = \alpha \int_{0}^{\theta} d\theta \]Angular displacement equation:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]where θ is angular displacement over time t.
Velocity-displacement equation:
\[ \omega^2 = \omega_0^2 + 2\alpha\theta \]which links angular velocity and displacement directly.
2. Acceleration: Linear and Angular
Just as we have two types of velocity, we also have two types of acceleration.
- Linear Acceleration (a
): This is the rate of change of an object’s linear velocity.
- a
=dtdv
- For a particle in uniform circular motion (constant speed, but changing direction), the linear acceleration, known as centripetal acceleration (ac), is directed towards the center of the circle. This is because the direction of the linear velocity is continuously changing. The magnitude is given by:ac=rv2=ω2r
- If the speed also changes, there’s an additional component called tangential acceleration (at), which is tangent to the circular path. The total linear acceleration is the vector sum of these two components:a
=ac
+at
- a
- Angular Acceleration (α
): This is the rate of change of an object’s angular velocity.
- α
=dtdω
- Its SI unit is radians per second squared (rad/s2). A non-zero angular acceleration means the object’s rotation is either speeding up or slowing down.
- α
Relation Between Linear and Angular Acceleration
The relationship between linear and angular acceleration is more nuanced and involves both centripetal and tangential components.
Let’s start with the vector form:
v=ω
×r
. Differentiating with respect to time (
t
):
dtdv=dtd(ω
×r
)
Using the product rule for differentiation of cross products:
dtdv=(dtdω
)×r
+ω
×(dtdr
)
Substituting the definitions of acceleration and velocity:
a=α
×r
+ω
×v
This is the general relation. Let’s analyze the two terms on the right side:
- Tangential Acceleration (at
): The term α
×r
represents the tangential acceleration. It’s the component of linear acceleration responsible for the change in the magnitude of the linear velocity (speeding up or slowing down). Its magnitude is:at=αr
- The direction of at
is tangential to the circular path.
- The direction of at
- Centripetal Acceleration (ac
): The term ω
×v
represents the centripetal acceleration. It’s the component of linear acceleration responsible for the change in the direction of the linear velocity. Its magnitude is:ac=ωv=ω(ωr)=ω2r=rv2
- The direction of ac
is always radially inward, towards the center of the circular path.
- The direction of ac
Therefore, the total linear acceleration is the vector sum:
a=at
+ac
=(α
×r
)+(ω
×v
)
The magnitude of the total linear acceleration is given by the Pythagorean theorem, since the centripetal and tangential components are perpendicular to each other:
a=at2+ac2=(αr)2+(ω2r)2
