Rotational Motion

Moment of Inertia:

Torque created by external forces in rotating motion.

When a particle is rotating it has tangential acceleration and radial acceleration.

Radial acceleration = ω²r

Hence radial force acting on it = m ω²r

Tangential acceleration = dv/dt = rdω/dt = r α
ω = angular velocity

α = angular acceleration

Tangential force = mrα

Torque created by radial force is zero as the force intersects the axis of rotation.

Tangential force and axis are skew (they do not intersect) and they are perpendicular. Thus the resultant torque is Force*radius = mr²α

In the case of a body having ‘n’ particles and rotating, the total torque is equal to torque acting on each of the particles

Total torque on the body = Γ(total) = Σ m_ir_i²α
Σ m_ir_i²α is called as moment of inertia.

Moment of inertia can be calculated using the above formula for collection of discrete particles.
For continuous bodies:

If the body is a continuous, the technique of integration needs to be used. We consider a small element of the body with mass dm and having a perpendicular distance from the axis or line about which moment of inertial is to be calculated.

We find ∫ r²dm under proper limits to get the moment of inertia of the body.

r²dm is the moment of inertia of the small element.

Moment of Inertia and Radius of Gyration for different bodies

BODY	axis of rotation	image	M.I	Radius of Gyration
uniform rod	perpendicular to its plane passing through its centre		ML2/12	L/2√3
uniform rod	perpendicular to its plane passing through its end		ML2/3	L/√3
Ring	perpendicular to its plane passing through its centre		MR2	R
Ring	along its diameter		MR2/2	R/√2
Ring	along a tangent in its plane		3MR2/2	√(3/2)R
Ring	along a tangent perpendicular to its plane		2MR2	√2R
Disc	perpendicular to its plane passing through its centre		MR2 /2	R/√2
Disc	along its diameter		MR2/4	R/2
Disc	along a tangent perpendicular to its plane		3MR2/2	√(3/2)R
Disc	along a tangent in its plane		5MR2/4	√5R/2
Hollow Sphere	along its diameter		2MR2/3	√(2/3)R
Solid Sphere	along its diameter		2/5MR2	√(2/5)R
Solid Sphere	along a tangent		7MR2/5	√(7/5)R
Solid Cylinder	about its own axis		MR2/2	R/√2
Solid Cylinder	about an axis passing through its centre of mass and perpendicular to its plane		M(L2/12+R2/4)	√(L2/12+R2/4)
Hollow Cylinder	about its own axis		MR2	R
Hollow Cylinder	about an axis passing through its centre of mass and perpendicular to its plane		M(L2/12+R2/2)	√(L2/12+R2/2)
Rectangular lamina	about an axis passing through its centre of mass and perpendicular to plane		M[(L2+b2)/12]	√[(L2+b2)/12]

Brain Teasers: Dynamics

1.A long thin uniform rod lies flat on the table as shown.One end of the rod is slowly pulled up by a force that remains perpendicular to the rod at all times. What minimum coefficient of static friction is required so that the rod can be brought to the vertical position without any slipping of the bottom end?

2.A small block slides down a slanted board when released. The upper half of the board is smooth and the lower is rough(each section is of equal length), so that the acceleration of the block on the smooth half is three times greater than it is on the rough half. The block reaches the bottom of the board in time t₁. The board is then flipped so that the upper half is rough and the lower part is smooth,and the block is released from the top again. This time,the block reaches the bottom of the board in time t₂. In both cases, the board makes the same angle with the horizontal. Find the ratio t₁/t₂.




3.Imagine that the mass of the Sun instantly doubles. How long would the Earth’s year be?

4.A dumbbell consists of a light rod of length r and two small masses m attached to it. The dumbbell stands vertically in the corner formed by two frictionless planes. After the bottom end is slightly moved to the right, the dumbbell begins to slide. Find the speed u of the bottom end at the moment the top end loses contact withthe vertical plane.






5.A sled is given a quick push up the snowy slope. The sled slides up and then comes back down; the whole trip takes time t. If the coefficient of sliding friction between the sled and the snow is µ, find the time t_u it took the sled to reach the top point of its trajectory. The slope makes the angle θ with the horizontal.

Rotational Kinematics & Dynamics

1. A uniform thin rod AB of mass 3m and length 2a is free to rotate in a horizontal plane about a smooth fixed vertical axis which passes through the midpoint O of the rod. Two small smooth rings each of mass m are free to slide on the rod. At time t=0 the rings are on the opposite side of O and are at a distance a/2 from O. The rod is then given an initial angular velocity 2√g/a, the rings being initially at rest relative to the rod.


i)Show that when the rings are about to slip off the rod its angular velocity is √g/a.
ii) Find the speed of either ring at this instant.
iii)If the point O is at a height a above the horizontal plane show that the distance between the points where the ring strikes the plane is 2a√(6 + 2√3).


2.Two particles m₁ and m₂ are connected by a light elastic string of natural length l₀ and are placed on a smooth horizontal table with separation between m₁ and m₂ as l₀.m₂is projected with a velocity vo along the table at right angles to the string. Show that the max length l of the particle is given by: l²(l-l₀)=m₁m₂vo2(l+l₀)/k(m₁+m₂) where k is force const of string .

3.Two equal particles are attached to the end of a light rod and a 3rd equal particle is connected by an inextensible string to a point on the rod at distance a and b from its ends. The latter particle is projected at right angles to the rod with a velocity v₀. Show that when the string becomes taut the velocity of the particle is changed to v₁=v₀(a²+b²)/2(a²+ab+b²) and that the rod begins to turn with angular velocity (a-b/a²+b²)v₁.






4.A rod of length 2a and mass M is in motion in a horizontal plane with speed u at right angle to its length when it collides with a small elastic sphere of equal mass whose center lies in the same plane. If the sphere is free to move, Prove that angular velocity acquired by the rod cannot exceed (1+e)u√6/4a.


5.A cube rests on a rough plane of inclination θ (< π/4) with 2 of its upper and two of its lower edges horizontal. A rope is attached to the midpoint of its upper most edge and is pulled parallel to the greatest slope of the plane. Show that it will be impossible to drag the cube without upsetting it if the coefficient of friction exceeds (1-tanθ )/2

6. A uniform rod of length 2a, mass m is set in motion by a sudden blow J acting at A and inclined at q to rod as shown. Find velocity of A of rod immediately after the impact and Prove that energy communicated is J²(1+3sin²θ)/2m.

7.A small block of mass m is placed inside a hollow cone rotating about a vertical axis with angular velocity ω as in the figure. The semivertical angle of the cone is θ and the coefficient of friction between the cone and the block is µ. If the block is to remain at a constant height h above the apex of the cone, what are the maximum and minimum values of ω?

8.The shown arrangement is in a horizontal plane. The particle with mass m is restricted to move in an elliptical orbit with shown dimension. The particle has velocity v at the shown position towards right. Length of rod is 2l. The coefficient of restitution between the particle and the rod is 1/3. The moment of inertia of rod is I, about O.


i) Find the locus of mid-point of spring of force constant K₃ .
ii) Find the frequency of the rod.
iii)What would be the maximum angular displacement of the oscillating system?
(Assume only one collision)

9.The system shown is initially at rest. Find the acceleration of the bobbins A and mass m. Assume that A can roll without slipping along the incline. Given that the MI about instantaneous axis of rotation of bobbins A,B,C is I_A, I_B, I_C. Assume their masses to be m_A, m_B, m_C and internal and external radii to be r and R.