Mathematics Conservation of angular momentum, EQUILIBRIUM OF A RIGID BODY AND Principle of moments

Topic covered

star Conservation of angular momentum
star EQUILIBRIUM OF A RIGID BODY
star Principle of moments

Conservation of angular momentum

● If tau_(ext)=0 Eq. (7.28b) reduces to

(dL)/(dt)=0

or color{blue}{L= "constant"........................(7.29a)}

● Thus, if the total external torque on a system of particles is zero, then the total angular momentum of the system is conserved, i.e. remains constant. Eq. (7.29a) is equivalent to three scalar equations,

color{blue}{L_x =K_1 , L_y =K_2" and "L_2 =K_3......................(7.29)}

● Here K_1, K_2 and K_3 are constants; L_x, L_y and L_z are the components of the total angular momentum vector L along the x,y and z axes respectively.
● The statement that the total angular momentum is conserved means that each of these three components is conserved. Eq. (7.29a) is the rotational analogue of Eq. (7.18a), i.e. the conservation law of the total linear momentum for a system of particles. Like Eq. (7.18a), it has applications in many practical situations.
Q 3230545412

Find the torque of a force 7 hat i + 3 hat j – 5 hat k  about the origin. The force acts on a particle whose position vector is
hat i – hat j + hat k

Solution:

Here r = hat i − hat j +hat k

and F= 7 hat i + 3 hat j - 5 hat k

We shall use the determinant rule to find the torque color{blue}{t = r xx F}

tau= |( hat i, hat j, hat k),(1,-1,1),(7, 3,-5)|

= (5-3) hat i -(-5-7) hat j +(3-(-7)) hat k

or tau = 2 hat i + 12 hat j + 10 hat k
Q 3240645513

Show that the angular momentum about any point of a single particle moving with constant velocity
remains constant throughout the motion.

Solution:

Let the particle with velocity v be at point P at some instant t. We want to calculate the angular momentum of the particle about an arbitrary point O.

The angular momentum is color{blue}{I = r xx mv}. Its magnitude is color{blue}{mvr sin theta}, where theta is the angle between r and v as shown in Fig. 7.19. Although the particle changes position with time, the line of direction of v remains the same and hence color{blue}{OM = r sin theta}. is a constant.

Further, the direction of I is perpendicular to the plane of r and v. It is into the page of the figure.This direction does not change with time. Thus, I remains the same in magnitude and direction and is therefore conserved. Is there any external torque on the particle?

EQUILIBRIUM OF A RIGID BODY

● We are now going to concentrate on the motion of rigid bodies rather than on the motion of general systems of particles.

● We shall recapitulate what effect the external forces have on a rigid body. (Henceforth we shall omit the adjective ‘external’ because unless stated otherwise, we shall deal with only external forces and torques.)
● The forces change the translational state of the motion of the rigid body, i.e. they change its total linear momentum in accordance with Eq. (7.17). But this is not the only effect the forces have.
● The total torque on the body may not vanish. Such a torque changes the rotational state of motion of the rigid body, i.e. it changes the total angular momentum of the body in accordance with Eq. (7.28 b).

● A rigid body is said to be in mechanical equilibrium, if both its linear momentum and angular momentum are not changing with time, or equivalently, the body has neither linear acceleration nor angular acceleration. This means

(1) the total force, i.e. the vector sum of the forces, on the rigid body is zero;

color{blue}{F_1+F_2+....F_n= sum_(i=1)^n F_i =0...........................(7.30 a)}

=> If the total force on the body is zero, then the total linear momentum of the body does not change with time. Eq. (7.30a) gives the condition for the translational equilibrium of the body.

(2) The total torque, i.e. the vector sum of the torques on the rigid body is zero,

color{blue}{tau_1+tau_2+.....tau_n= sum_(i=1)^n tau_i=0.................(7.30 b)}

● If the total torque on the rigid body is zero, the total angular momentum of the body does not change with time. Eq. (7.30 b) gives the condition for the rotational equilibrium of the body.

● One may raise a question, whether the rotational equilibrium condition [Eq. 7.30(b)] remains valid, if the origin with respect to which the torques are taken is shifted. One can show that if the translational equilibrium condition [Eq. 7.30(a)] holds for a rigid body, then such a shift of origin does not matter, i.e. the rotational equilibrium condition is independent of the location of the origin about which the torques are taken.
● Example 7.7 gives a proof of this result in a special case of a couple, i.e. two forces acting on a rigid body in translational equilibrium. The generalisation of this result to n forces is left as an exercise.

Eq. (7.30a) and Eq. (7.30b), both, are vector equations. They are equivalent to three scalar equations each. Eq. (7.30a) corresponds to

color{blue}{sum_(i=1)^n F_(ix) =0 , sum_(i=1)^n F_(iy)=0 " and " sum_(i=1)^n F_(iz)=0....................(7.31 a)}

=> where F_(ix), F_(iy) and F_(iz) are respectively the x, y and z components of the forces F_i. Similarly, Eq. (7.30b) is equivalent to three scalar equations

color{blue}{sum_(i=1)^n tau_(ix)=0, sum_(i=1)^n tau_(iy)=0 " and " sum_(i=1)^n tau_(iz)=0.................(7.31 b)}

where τ _(ix), τ _(iy) and τ _(iz) are respectively the x, y and z components of the torque τ_ i .

=> Eq. (7.31a) and (7.31b) give six independent conditions to be satisfied for mechanical equilibrium of a rigid body.
● In a number of problems all the forces acting on the body are coplanar. Then we need only three conditions to be satisfied for mechanical equilibrium.
● Two of these conditions correspond to translational equilibrium; the sum of the components of the forces along any two perpendicular axes in the plane must be zero. The third condition corresponds to rotational equilibrium. The sum of the components of the torques along any axis perpendicular to the plane of the forces must be zero.

● The conditions of equilibrium of a rigid body may be compared with those for a particle, which we considered in earlier chapters. Since consideration of rotational motion does not apply to a particle, only the conditions for translational equilibrium (Eq. 7.30 a) apply to a particle.
● Thus, for equilibrium of a particle the vector sum of all the forces on it must be zero. Since all these forces act on the single particle, they must be concurrent. Equilibrium under concurrent forces was discussed in the earlier chapters.

● A body may be in partial equilibrium, i.e., it may be in translational equilibrium and not in rotational equilibrium, or it may be in rotational equilibrium and not in translational equilibrium.

● Consider a light (i.e. of negligible mass) rod (AB), at the two ends (A and B) of which two parallel forces both equal in magnitude are applied perpendicular to the rod as shown in Fig. 7.20(a).

● Let C be the midpoint of AB, CA = CB = a. the moment of the forces at A and B will both be equal in magnitude (aF), but opposite in sense as shown.

colo{blue}☞ The net moment on the rod will be zero. The system will be in rotational equilibrium, but it will not be in translational equilibrium; sum F ne 0

● The force at B in Fig. 7.20(a) is reversed in Fig. 7.20(b). Thus, we have the same rod with two equal and opposite forces applied perpendicular to the rod, one at end A and the other at end B. Here the moments of both the forces are equal, but they are not opposite, So The total force on the body is zero; so the body is in translational equilibrium; but it is not in rotational equilibrium.

\color{red} ✍️ A pair of equal and opposite forces with different lines of action is known as a couple. A couple produces rotation without translation.

● When we open the lid of a bottle by turning it, our fingers are applying a couple to the lid [Fig. 7.21(a)]. Another known example is a compass needle in the earth’s magnetic field as shown in the Fig. 7.21(b).
● The earth’s magnetic field exerts equal forces on the north and south poles. The force on the North Pole is towards the north, and the force on the South Pole is toward the south. Except when the needle points in the north-south direction; the two forces do not have the same line of action. Thus there is a couple acting on the needle due to the earth’s magnetic field.

Q 3250156014

Show that moment of a couple does not depend on the point about which you take the moments.

Solution:

Consider a couple as shown in Fig. 7.22 acting on a rigid body. The forces F and -F act respectively at points B and A. These points have
position vectors r_1 and r_2 with respect to origin O. Let us take the moments of the forces about
the origin.

The moment of the couple = sum of the moments of the two forces making the couple

color{blue}{=r_1 xx (-F)+r_2 xx F}

=r_2 xx F -r_1 xx F

=(r_2 -r_1) xx F

But color{blue}{r_1 + AB = r_2}, and hence color{blue}{AB = r_2 – r_1}. The moment of the couple, therefore, is color{blue}{AB xx F}.

Clearly this is independent of the origin, the point about which we took the moments of the forces.

Principle of moments

● An ideal lever is essentially a light (i.e. of negligible mass) rod pivoted at a point along its length. This point is called the fulcrum.
● A seesaw on the children’s playground is a typical example of a lever. Two forces F_1 and F_2, parallel to each other and usually perpendicular to the lever, as shown here, act on the lever at distances d1 and d2 respectively from the fulcrum as shown in Fig. 7.23.

● The lever is a system in mechanical equilibrium. Let R be the reaction of the support at the fulcrum; R is directed opposite to the forces F_1 and F_2. For translational equilibrium,

color{blue}{R-F_1-F_2=0.....................(i)}

● For considering rotational equilibrium we take the moments about the fulcrum; the sum of moments must be zero,

color{blue}{d_1F_1-d_2F_2=0...................(ii)}

● Normally the anticlockwise (clockwise) moments are taken to be positive (negative). Note R acts at the fulcrum itself and has zero moment about the fulcrum.

● In the case of the lever force F_1 is usually some weight to be lifted. It is called the load and its distance from the fulcrum d_1 is called the load arm. Force F_2 is the effort applied to lift the load; distance d_2 of the effort from the fulcrum is the effort arm.

Eq. (ii) can be written as

color{blue}{d_1F_1=d_2F_2.........................(7.32 a)}

or load arm xx load = effort arm xx effort The above equation expresses the principle of moments for a lever. Incidentally the ratio F_1//F_2 is called the Mechanical Advantage (M.A.);

color{blue}{M.A=(F_1)/(F_2)= d_2/d_1........................(7.32)}

● If the effort arm d_2 is larger than the load arm, the mechanical advantage is greater than one. Mechanical advantage greater than one means that a small effort can be used to lift a large load.
● There are several examples of a lever around you besides the see-saw. The beam of a balance is a lever. Try to find more such examples and identify the fulcrum, the effort and effort arm, and the load and the load arm of the lever in each case.

You may easily show that the principle of moment holds even when the parallel forces F_1 and F_2 are not perpendicular, but act at some angle, to the lever.