The Rotational motion for better understanding.
Most of us are familiar with rotational motion in our day-to-day life. Rotational motion is the motion of a body around a circular path in a fixed orbit, where only rigid bodies are considered. A rigid body is a body with a definite perfect unchanging shape. In such a body the distance between all pairs of particles doesn’t change.
What is rotational motion, definition;
Rotational motion is the motion of a body around a circular path in a fixed orbit, where only rigid bodies are considered. A rigid body is a body with a definite and perfect unchanging shape. In such a body the distance between all pairs of particles doesn’t change and remains constant all the time.
Rotational motion about a fixed axis;
A rigid body can undergo two types of motions, such as translational motion and rotational motion. However, in rotational motion, the rigid body has to rotate about a fixed axis, where every particle of the body moves in a circle, which lies in a plane perpendicular to the axis having its center on the axis.
Let us see the above image A, of a rigid body moving around a fixed axis Z. Suppose P1 is a particle of the rigid body placed at distance r1 from the center C1 on the fixed axis. The circle lies in a plane perpendicular to the axis z. Similarly, another particle P2 of the rigid body is at a distance of r2 from the fixed axis with a center at C2. This bigger circle also lies in a plane perpendicular to the axis.
It is to be noted here that in spite of placing the two circles in two different planes, both are perpendicular to the fixed axis z. For P3, r is zero, hence no circle is formed.
Examples of rotational motion;
- Ceiling fan: When a ceiling fan is on every part of its body goes in a circle. Here all the blades of the fan rotate in the same magnitude about a fixed vertical line /axis.
- Switch of a gas stove: Everyone is familiar with the operation of a gas stove. To put the gas on one has to push the gas switch a little bit and rotate left and put off to rotate right. At the time of rotation, each particle of the switch moves in a circle about a fixed axis.
- A potter’s wheel: A potter’s wheel is used to make different types of earthenware. When the wheel rotates every part of the wheel moves in a circle about a fixed axis in a plane perpendicular to its axis.
In the Kinemetic of rotational motion relationship between the angle of rotation, angular velocity, angular acceleration, and time is described.
Let us go through image B, where a rigid body rotates about a fixed axis z. The particle describes as a circle with a center at C and having radius r, which is the perpendicular distance of point P from the axis. When the particle is positioned at P1, it covers angular displacement ∆θ in a time interval of ∆t. The equation is given below in angular velocity.
The change of angular displacement with time is called the angular velocity of the particle. Let us assume that the angular position of the body at t is θ. Now the average angular velocity of the body can be written as,
- ∆θ is the change in angular displacement.
- ∆t is a time interval.
Now equation of instantaneous angular velocity at time t is
We know about angular displacement and angular velocity ω in rotational motion. If the body rotates through an equal angular distance in equal intervals of time (irrespective of the smallness of the interval) it is said uniform angular velocity of the body. In uniform angular velocity, the acceleration is zero. In such case ω=dθ/dt=constant.
Hence θ= ωt.
In rotational motion, the angular acceleration is analogous to linear acceleration, and angular acceleration α may be defined as the time rate of change of angular velocity, and the equation is written as;
- Where α is angular acceleration.
- dω is the change in angular velocity.
- dt is a time interval.
- SI units are rad/sec2
Dynamics of rotational motion;
It is needless to mention here that the dynamics of rotational motion are entirely analogous to linear or translational dynamics. Even many of the equations of both rotational and linear motions are similar. It is a fact that in rotational motion moment of inertia and torque play the same role which corresponds to the mass and force of linear motion respectively.
The rotational analog of force is the moment of force or torque. In other words, it may be said as the angular version of force. The dimensions of torque are written as ML2T-2. Its dimension is the same as those of work or energy. Torque is a vector quantity whereas work is a scalar.SI unit of torque is N-m (Newton meter).
The equation of torque is;
Ʈ=r x F, where
- Ʈ is torque perpendicular to the plane having r and F.
- r is the position vector with respect to its origin.
- F is the force applied to it.
The magnitude of the moment of force or torque may be written as Ʈ=(rsinθ) F=r1F
Or Ʈ=rF sinθ=rF1, where
- r1=rsinθ is the perpendicular distance of the line of action of F from the origin
- F1(=F sinθ) is the component of F in the direction perpendicular to r.
It is to mention here that r x F, is a vector product and properties of two vectors are applied to it. When the direction of F is reversed, the direction of the torque is also reversed. However, if the directions of both r and F are reversed, the direction of torque remains the same.
As we are aware of the fact that the rotational analog of force is torque, similarly the angular momentum is the rotational analog of linear momentum. Angular momentum is a vector quantity. It can be referred to as a moment of (linear) momentum.
Suppose a particle of mass m and linear momentum p at r relative to the origin O. The angular momentum L of the particle with respect to the origin O is defined as L=r x p
The magnitude of the angular momentum vector is l=rp sinθ, where
- p is the magnitude of p
- θ is the angle between r and p
we may have the following equation
l=r p1 or r1 p where
- r1(=rsinθ) is the perpendicular distance of the directional line of p from the origin.
- P1(=psinθ) is the component of p in a direction perpendicular to r.
The angular momentum shall be zero(l=0), if the particle is at the origin(r=0) or if the directional line of p passes through the origin θ=0 degree or 180 degrees.
Conservation of Angular momentum;
Conservation of angular momentum is a physical property of a spinning system, that it continues spinning unless it is acted upon by an external torque. It is determined by an object’s mass and velocity. When net torque is zero, the angular momentum remains constant, which means the momentum remains conserved.
Angular momentum is the rotational analog of linear momentum. The equation of angular momentum of a particle in a rotating motion can be,
l=r x p
- r=is the radius of the circle formed by the body in rotational motion.
- P=linear momentum of the body.
And the magnitude of the angular momentum will be,
l=r p sin θ
Moment of inertia;
Inertia is a property of a body, which is responsible for resistance to any change in its present state of rest or motion. In rotational motion about a fixed axis, the moment of inertia plays an important role as mass does in linear motion. It is often referred to as angular mass or rotational inertia.
The moment of inertia of an object is a determined measurement of a rigid body rotating around a fixed axis. Here axis is always considered as a parameter to describe the relationship in the moment of inertia.
The formula of the moment of inertia;
The formula for the moment of inertia is as follows;
- m is the sum of the product of the mass.
- r is the distance from the axis of rotation.
Comparison of Translational and Rotational Motion.
The relation between linear or translational motion and rotational motion is always referred to whenever the rotational motion is discussed. Because different parameters involved in rotational motion are only the rotational versions of linear motion. The following table describes the facts.
Rotational motion is one of the major types of motion. Many of the parameters are analogous to linear motion and a subset of circular motion. For a better understanding of rotational motion, an understanding of linear motion is required.