Rotational motion is experienced by rigid bodies as well as translational motion. Therefore, the linear and angular velocities need to be analyzed in such cases. This problem can be simplified by separating the translational and rotational motion of the body. This article will talk about how an object rotates around a fixed axis.
An object's rotation can be defined as its motion around a circle in a fixed orbit.
Rotational Motion is entirely analogous to translational or linear motion. Several of the equations for the mechanics of rotating objects are also applicable to linear motion. However, only rigid bodies are considered in rotational motion. An object with a fixed mass and shape is considered rigid.
Whenever a rigid body undergoes a rotation around a fixed axis, we only need to consider forces acting in planes perpendicular to that axis.
Example of Rotation around a Fixed Point The rotation of the ceiling fan blades, rotation of the minute hand and hour hand in the clock, as well as opening and closing the door, are a few examples of rotation about a fixed point.
Example of rotation around an axis
The motion of a rotating axis includes both translational and rotational motion.
For example, pushing a ball from an inclined plane is an excellent example of
rotation about an axis of rotation. During translation, the ball reaches the
bottom of the inclined plane, and the ball's motion occurs when it rotates about
its axis, which is rotational motion.
Earth's motion is also an example of rotation about an axis of rotation. In
addition to rotating around its axis every day, the earth also revolves around
the sun once a year. Translation and rotation are displayed in this classic
example of motion.
Moment of Inertia
As the rotation of an object changes, the moment of inertia also changes.
The symbol symbolizes the moment of inertia I, measured in kilogram
metre2 (kg m2). According to the following equations, the
moment of inertia is:
I = Mr2, where m is the particle's mass and r is the
particle's distance from the axis of rotation. A particle's moment of inertia is
determined by its mass; the larger the mass, the greater the moment of inertia.
Torque
Torque can be defined as the twisting effect of a force applied to a rotating
object r degrees away from its axis of rotation. On a mathematical level, this
relationship looks like this:
τ = rxF
Angular Momentum
An object's angular momentum L measures the difficulty of bringing it to rest
after rotating. It can be calculated as follows:
L = ∑ rxp
Work-energy theory states that the total work done by all forces acting on a system will equal the change in kinetic energy. In the concept of work energy, torque is used to describe rotational motion. When a force is applied, an object is balanced if its displacements and rotations account for zero work.
The object rotates in a small amount when it is a rigid body. Then the linear displacement is calculated as Δr = rΔθ. It is perpendicular to r.
As a result, the work done is
ΔW = F perpendicular to Δr
ΔW = F Δr sin φ
ΔW = Fr Δθ sin φ
ΔW = τΔθ
Increasing the number of forces acting will increase the work done
ΔW = (τ1 + τ2 + ......) Δθ
However, we know that all forces are the same. Therefore, there will be no work done, which is
τ1 + τ2 + ...... = θ
Thus, rotational motion can be explained by the work-energy principle.
Relations among torque, a moment of inertia, and angle of acceleration
Those who have ever pushed a merry-go-round can understand the rotational dynamics; the angular velocity changes when a force is applied to a merry-go-round. Likewise, bicycle wheels also spin when force is used as the force increases, the angular acceleration produced in the wheel increases. As a result, we can say that there is a relationship between forces, masses, angular velocity, and angular acceleration.
Take a look at the wheel of the bike. A wheel's acceleration, for example, is the result of angular forces F acting on it. R is the radius of the wheel. We know the force acts perpendicularly to the radius. In addition, we know that,
F = ma
Where a is acceleration = rα
Therefore,
F = mrα
As we have learned, torque is the effect of force turning. Thus,
τ = Fr
rF = mr2α
τ= mr2 α
We may therefore say that the last equation is the rotational analog of F = ma, such that torque is equivalent to force, angular acceleration is equivalent to acceleration, and rotational inertia is equal to mass. As well as rotational inertia, a moment of inertia is also commonly used.
Angular acceleration and torque are related by the moment of inertia
net τ = Iα
α = net τ/I
Where net τ is the total torque.
