2. rotational motion

Angular velocity and linear velocity

Consider a rigid body rotating with about a fixed axis (the Z axis) which is perpendicular to the plane of the paper. Now consider a point P at a distance ‘r’ from the origin.  As the rigid body rotates the particle at point P moves with a constant speed ‘v’ in a circular path as shown by the dotted line. Here ‘v ‘is the linear velocity of the particle at a distance ‘r’ from the origin

Suppose the particle P moves a linear displacement and angular displacement of ‘Δθ' in time ‘dt’, angular velocity  w=Δθ/dt

If ‘r’ is the distance of particle P from the origin. Then   s=rθ .    linear velocity  v= ds/dt  = ddt(rθ) =r ddtθ = r w  

So we have the relation between linear and angular velocityv= r w. 

Important results from the relation

• The linear velocity is different at different points on the circle.
• It is zero at the center. It is a minimum in between the center and any point on the circumference.
• It is maximum at the circumference of the circle.
• However, angular velocity remains the same at all points on the circular path. 

We refer to ω as the angular velocity of the whole body.

We have characterized pure translation of a body by all parts of the body having the same velocity at any instant of time. Similarly, we may characterize pure rotation by all parts of the body having the same angular velocity at any instant of time.

Angular displacement is considered as an axial vector: We know that angular displacement is the angle traced by a particle under circular motion. Since the direction of the displacement is along the axis, that’s why angular displacement is an axial vector.

Similar to the displacement, the angular velocity is also an axial vector quantity. We can find its direction by using the Right-hand Thumb Rule. In general, we have the relation  v= w ×r

This rule says: Curl your fingers in a counterclockwise direction, and the thumb pointing outwards (along the axis) is the direction of the angular velocity. Similarly, if you curl your fingers in a clockwise direction, then the thumb pointing inwards gives the direction of ω.

Angular acceleration

In rotational motion, the concept of angular acceleration in analogy with linear acceleration is defined as the time rate of change of velocity in translational motion.

Angular acceleration, also called rotational acceleration, is a quantitative expression of the change in angular velocity that a spinning object undergoes per unit of time. It is a vector quantity, consisting of a magnitude component and either of two defined directions or senses.  

The vector direction of the acceleration is perpendicular to the plane where the rotation takes place.  Increase in angular velocity clockwise, then the angular acceleration velocity points away from the observer. If the increase in angular velocity is counterclockwise, then the vector of angular acceleration points toward the viewer.

Relation between linear acceleration and angular acceleration

Which finally gives the relation   a=r α, as dw/dt=α, In vector form  a= α×r

Moment of force: Torque

We have learned that the motion of a rigid body is a combination of both translational and rotational motion. But if we fix the rigid body by one point or line, it can do only rotational motion.

In translational motion, an external force is required for the translational motion to happen (to produce a linear acceleration in the body). What must be the rotational analog of force?  What makes a rigid body rotate?

Take an example of a door. A door is a rigid body that can rotate about a fixed vertical axis passing through the hinges. What makes the door rotate? It is clear that unless a force has been applied the door does not rotate. But alone force cannot do the job.

If we apply the force on the hinge, the door will not rotate. But if we apply force at right angles to the door at its outer edge it is most effective in producing rotation. It is not the force alone, but how and where the force is applied is important in rotational motion.

The rotational analog of force is the Moment of force. The turning effect of a force is known as the moment of force, which is also known as torque.

It is the product of the force multiplied by the perpendicular distance from the line of action of the force to the pivot or point where the object will turn.

torque τ = F r; Here F= force and r= perpendicular distance of force from the axis of rotation

 In vector form, Moment of force  ( Torque)  τ = r × F

The magnitude of Torque  τ= r F sinθ; where θ is the angle between  r and F

The SI unit of moment of a force is Newton-meter (Nm). It is a vector quantity.

Torque will be minimum when θ = 0, 180   Then τ= rF sin 0=0

Torque will be maximum when θ=90,  Then   τ= r F sin 90 = rF

 Since r × F is a vector product, properties of a vector product of two vectors apply to it.

Angular momentum

In translational motion, we have linear momentum which is the product of mass and linear velocity P= mv and we have seen the concept of conservation of linear momentum when a net external force is applied to the system is zero.  What could be the rotational analog of linear momentum?

The rotational analog of linear momentum is angular momentum. Just like the moment of force is a vector product of r and F. The angular momentum is also a vector product of r and P (linear momentum)

Since angular momentum is the vector product of r and P, it is also a vector quantity and follows all the properties of the vector product.

L would be zero when θ=0 , 180  ; L= rF sin 0= 0

 How to calculate L as a vector using r and F in vector form?

Suppose we want to calculate  L= lx i + l y j + lz k And we are given position vector r and linear momentum P in vector form as given below then,

The physical quantities, a moment of force and angular momentum, have an important relationship between them. It is the rotational analog of the relation between force and linear momentum

Let us derive the relation between the moment of force and the angular momentum,

Thus, the time rate of change of the angular momentum of a particle is equal to the torque acting on it. This is the rotational analog of the equation F = dp/dt, which expresses Newton’s second law for the translational motion of a single particle

Moment of force and angular momentum

When no external force is applied on a system of particles, so from Newton’s second law dP/dt = Fext=0  as  Fext= 0. 

This relation suggests that when the external force on the system of particles is zero then its linear momentum is conserved.

We have rotational analog of this relation as  d dt L= τ ext,

Thus the time rate of the total angular momentum of a system of particles about a point is equal to the sum of the external torques acting on the system of particles taken about the same point.

Conservation of angular momentum

Or we can say that L= constant.

Thus, if the total external torque acting on a system is zero. Then the total angular momentum of the system is conserved or remains constant. This is called the conservation of angular momentum.

This is the rotational analog of conservation of linear momentum in translational motion.

Equilibrium of rigid body

Equilibrium is a state of the body where neither the internal energy nor the motion of the body changes with respect to time. Let us try to understand the equilibrium of a rigid body.

If we have to define equilibrium the simplest definition would be it is a point where the net external force, as well as torque acting on the body about COM or any other point, is zero. But to be more specific for a rigid body equilibrium means both rotational and translational equilibrium. For example, consider the following situation:

In the mechanical equilibrium of a rigid body, the linear momentum and angular momentum remain unchanged with time. This implies that the body under the influence of external force neither has a linear acceleration nor an angular acceleration. We, therefore, can say that:

• If the total force on a rigid body is zero then the body shows translational equilibrium as the linear momentum remains unchanged despite the change in time.
• If the total torque on a rigid body is zero then the body shows rotational equilibrium as the angular momentum does not change with time.

 Mechanical equilibrium when we sum up the above findings of translational and rotational equilibrium we get the following relations.     F1 + F2 +F3 + F4  +.....+ Fn = i=1nFi =0 , Net force =0,  Also  τ1+ τ2+τ3+τ4+....+ τn = i=1n τ i =0 , Net torque =0 , These equations are the vector in nature. As scalars the force and torque in their x, y, and z components are - Fi x=0 , Fiy=0 , Fiz=0 , also τ ix=0 , τiy=0 , τiz=0    . The independent condition of force and torque helps in reaching the rigid bodies to a state of mechanical equilibrium. Generally, the forces acting on the rigid body are coplanar. These conditions, if satisfied, help the rigid body attain equilibrium.