`●` Figure 7.34 shows a cross-section of a rigid body rotating about a fixed axis, which is taken as the z-axis (perpendicular to the plane of the page; see Fig. 7.33).
`●` Let `F_1` be one such typical force acting as shown on a particle of the body at point `P_1` with its line of action in a plane perpendicular to the axis.
`●` For convenience we call this to be the `x′ –y′` plane (coincident with the plane of the page). The particle at `P_1` describes a circular path of radius `r_1` with centre C on the axis; `CP_1 = r_1`.

`●` In time `Δt`, the point moves to the position `P_1′` . The displacement of the particle `ds_1`, therefore, has magnitude `ds_1 = r_1dθ` and direction tangential at `P_1` to the circular path as shown. Here `dθ` is the angular displacement of the particle, `dθ = ∠P_1 CP_ ′`

`\color{red} ✍️` The work done by the force on the particle is
`dW_1 = F_1* ds_1= F_1ds_1 cosφ_1= F_1(r_1 dθ)sinα_1`

`=>` where `φ_1` is the angle between `F_1` and the tangent at `P_1`, and `α_1` is the angle between `F_1` and the radius vector `OP_1; φ_1 + α_1 = 90°` .

`●` The torque due to `F_1` about the origin is `OP_1 xx F_1`. Now `OP_1 = OC + OP_1`. [Refer to Fig. 7.17(b).] Since `OC` is along the axis, the torque resulting from it is excluded from our consideration. The effective torque due to `F_1` is `τ_1= CP × F_1`; it is directed along the axis of rotation and has a magnitude `τ_1= r_1F_1 sinα` , Therefore,

` dW_1 = τ _1dθ` If there are more than one forces acting on the body, the work done by all of them can be added to give the total work done on the body. Denoting the magnitudes of the torques due to the different forces as `τ_ 1, τ_ 2, … `etc,

`color{blue}{dW = (τ_1 +τ_2 + ...)dθ}`


`bbul"Remember"`, the forces giving rise to the torques act on different particles, but the angular displacement `dθ` is the same for all particles. Since all the torques considered are parallel to the fixed axis, the magnitude `τ` of the total torque is just the algebraic sum of the magnitudes of the torques, i.e., `τ = τ_1 + τ_2 + .....` We, therefore, have

`color{blue} {dW=tau d theta.................(7.41)}`

`●` This expression gives the work done by the total (external) torque `τ` which acts on the body rotating about a fixed axis. Its similarity with the corresponding expression

`dW= F ds` for linear (translational) motion is obvious.

Dividing both sides of Eq. (7.41) by dt gives

`P=(dW)/(dt)= tau (d theta)/(dt)= tau omega`

or `P= tau omega.................(7.42)}`

`●` This is the instantaneous power. Compare this expression for power in the case of rotational motion about a fixed axis with the expression for power in the case of linear motion,

` P = Fv`

`●` In a perfectly rigid body there is no internal motion. The work done by external torques is therefore, not dissipated and goes on to increase the kinetic energy of the body. The rate at which work is done on the body is given by Eq. (7.42). This is to be equated to the rate at which kinetic energy increases. The rate of increase of kinetic energy is

`d/(dt) ((I omega^2)/2) =I ((2 omega))/2 (d omega)/(dt)`

`●` We assume that the moment of inertia does not change with time. This means that the mass of the body does not change, the body remains rigid and also the axis does not change its position with respect to the body.

Since `α = dω //dt`, we get

`d/(dt) ((I omega^2)/2)=I omega alpha`

`●` Equating rates of work done and of increase in kinetic energy,

`tau omega= I omega alpha`

`color{blue}{tau=I alpha.......................(7.43)}`

`●` Eq. (7.43) is similar to Newton’s second law for linear motion expressed symbolically as

`color{blue}{F = ma}`

`●` Just as force produces acceleration, torque produces angular acceleration in a body. The angular acceleration is directly proportional to the applied torque and is inversely proportional to the moment of inertia of the body. Eq.(7.43) can be called Newton’s second law for rotation about a fixed axis.

`●` As we know that the time rate of total angular momentum of a system of particles about a point is equal to the total external torque on the system taken about the same point.
When the total external torque is zero, the total angular momentum of the system is conserved.

`colo{blue}☞` We now wish to study the angular momentum in the special case of rotation about a fixed axis. The general expression for the total angular momentum of the system is

`color{blue}{L= sum_(i=1)^N r_i xx p_i.................(7.25)}`

`●` We first consider the angular momentum of a typical particle of the rotating rigid body. We then sum up the contributions of individual particles to get `L` of the whole body.

`●` For a typical particle `I = r xx p`. As seen in the last section `r = OP = OC + CP` [Fig. 7.17(b)]. With `p = m v`

`I=(OC xx mv)+(CP xx mv)`

`●` The magnitude of the linear velocity `v` of the particle at P is given by `v = wr_(bot)` where `r_(bot)` is the length of `CP` or the perpendicular distance of `P ` from the axis of rotation. Further, `v` is tangential at `P` to the circle which the particle describes.
`●` Using the right-hand rule one can check that `CP xx v` is parallel to the fixed axis. The unit vector along the fixed axis (chosen as the z-axis) is `hat k` . Hence

`CP xx mv= r_(bot)(mv) hat k`

`= m_(bot)^2 omega hat k` (since `v omega r_(bot) =` )

`●` Similarly, we can check that `OC xx v` is perpendicular to the fixed axis. Let us denote the part of l along the fixed axis (i.e. the z-axis) by `l_z`, then

`I_z= CP xx mv=mr_(bot)^2 omega hat k`

and `I=I_z+ OC xx mv`

`●` We note that `l_z` is parallel to the fixed axis, but `I` is not. In general, for a particle, the angular momentum l is not along the axis of rotation, i.e. for a particle, l and w are not necessarily parallel. Compare this with the corresponding fact in translation. For a particle, p and v are always parallel to each other.

`●` For computing the total angular momentum of the whole rigid body, we add up the contribution of each particle of the body.

Thus `L= sum I_i =sum l_(iz) + sum OC_i xx m_i v_i`

`●` We denote by `L_(⊥)` and ` L_(z)` the components of `L` respectively perpendicular to the z-axis and along the z-axis;

`color{blue}{L_(bot)= sum OC_i xx m_i v_i.................(7.44 a)}`

`=>` where `m_i` and `v_i` are respectively the mass and the velocity of the `i^(th)` particle and `C_i` is the centre of the circle described by the particle;

and `L_z = sum I_(iz) =( sum m_i r_i^2) omega hat k`

or `color{blue}{L_z= I omega hat k............(7.44 b)}`

`●` The last step follows since the perpendicular distance of the `i^(th)` particle from the axis is `r_i`; and by definition the moment of inertia of the body about the axis of rotation is `I= sum m_i r_i^2`

Note `color{blue}{L=L_z+L_(bot)....................(7.44 c)}`

`●` The rigid bodies which we have mainly considered in this chapter are symmetric about the axis of rotation, i.e. the axis of rotation is one of their symmetry axes.
`●` For such bodies, for a given `OC_i`, for every particle which has a velocity `v_i` , there is another particle of velocity `–v_i` located diametrically opposite on the circle with centre `C_i` described by the particle. Together such pairs will contribute zero to ` L_(bot)` and as a result for symmetric bodies `L_(bot)` is zero, and hence

`color{blue}{L=L_z=I omega hat k...............(7.44 d)}`

`●` For bodies, which are not symmetric about the axis of rotation, `L` is not equal to `L_z` and hence L does not lie along the axis of rotation. Referring to table 7.1, can you tell in which cases `L = L_z` will not apply?

`●` Let us differentiate Eq. (7.44b). Since `hat(k)` is a fixed (constant) vector, we get

`d/(dt)(L_z) =d/(dt) (I omega) hat k`

Now, Eq. (7.28b) states

`(dL)/(dt) = tau`

`●` As we have seen in the last section, only those components of the external torques which are along the axis of rotation, need to be taken into account, when we discuss rotation about a fixed axis.
`●` This means we can take `tau = tau hat k` . Since ` L = L_z + L_(bot)` and the direction of `L_z` (vector `hat k` ) is fixed, it follows that for rotation about a fixed axis,

`color{blue}{(dL_z)/(dt)= tau hat k..............(7.45 a)}`

and `color{blue}{(dL_(bot))/(dt)=0................(7.45 b)}`

`●` Thus, for rotation about a fixed axis, the component of angular momentum perpendicular to the fixed axis is constant. As ` L_z = Iomega hat k` , we get from Eq. (7.45a),

`color{blue}{d/(dt) (I omega)=tau....................(7.45 c)}`

If the moment of inertia `I` does not change with time,

`d/(dt)(I omega) =I (d omega)/(dt) = I alpha`

and we get from Eq. (7.45c),

`color{blue}{tau = I alpha.....................(7.43)}`

We have already derived this equation using the work - kinetic energy route.

`●` We are now in a position to revisit the principle of conservation of angular momentum in the context of rotation about a fixed axis. From Eq. (7.45c), if the external torque is zero,

`color{blue}{L_z = I omega = "constant" ....................... (7.46)}`

`=>` For symmetric bodies, from Eq. (7.44d), `L_z` may be replaced by `L` .(`L` and `L_z` are respectively the magnitudes of `L` and `L_z`.)

`●` This then is the required form, for fixed axis rotation, of Eq. (7.29a), which expresses the general law of conservation of angular momentum of a system of particles.
`●` Eq. (7.46) applies to many situations that we come across in daily life. You may do this experiment with your friend. Sit on a swivel chair with your arms folded and feet not resting on, i.e., away from, the ground. Ask your friend to rotate the chair rapidly. While the chair is rotating with considerable angular speed stretch your arms horizontally.



`●` What happens? Your angular speed is reduced. If you bring back your arms closer to your body, the angular speed increases again. This is a situation where the principle of conservation of angular momentum is applicable.

`●` If friction in the rotational mechanism is neglected, there is no external torque about the axis of rotation of the chair and hence `Iω` is constant. Stretching the arms increases `I` about the axis of rotation, resulting in decreasing the angular speed `ω` . Bringing the arms closer to the body has the opposite effect.
`●` A circus acrobat and a diver take advantage of this principle. Also, skaters and classical, Indian or western, dancers performing a pirouette on the toes of one foot display ‘mastery’ over this principle. Can you explain?

`●` A circus acrobat and a diver take advantage of this principle. Also, skaters and classical, Indian or western, dancers performing a pirouette on the toes of one foot display ‘mastery’ over this principle.