Comparison of moment of inertia with mass
In linear motion, greater in the mass of the body, greater is the force required to produce the linear acceleration in it. Thus in linear motion mass of the body is a measure of its inertia. In rotational motion a torque is applied to produce angular acceleration. Moment of inertia is a measure of rotational inertia of the body. In rotation the moment of Inertia plays a similar role as mass does in the linear motion.
A fly wheel is a circular disc, whose most of the mass is concentrated on its rim and it rotates about an axel passing through its centre and perpendicular to its plane. The machines such as steam engine and automobile engine that produce rotational motion have a fly wheel.
Working of Fly wheel
Jerky motion of a vehicle can be prevented by attaching a fly wheel with its engine. Since the most of the mass of the fly wheel is concentrated at its rim, the fly wheel has large moment of inertia. Therefore fly wheel opposes or resists the rotational motion to a great extent. Whenever there is a sudden increase or decrease in the speed of vehicle, the fly wheel opposes this sudden increase or decrease in the speed of vehicle due to its large moment of Inertia.
Factors on which moment of inertia depends
Moment of inertia depends on, (i) Position of the axis of rotation with respect to the body. (ii) Orientation of the axis of rotation. (iii) Mass of the rotating body. (iv) Distance from the axis of rotation to the distribution of mass.
Radius of gyration
It the distance of a mass point from the axis of rotation, whose mass is equal to the mass of the whole body and whose moment of inertia is equal to the moment of inertia of the whole body about the axis.
Explanation: The moment of inertia of the body about the axis of rotation is given by,
If m is the mass of the each particle,then,
𝐼 = 𝑚(𝑟12 + 𝑟22 + 𝑟32+. … … … + 𝑟𝑛2)
If the body contains n particles, then by multiplying and dividing RHS by n,
𝐼 = 𝑚𝑛 [(𝑟12 + 𝑟22 + 𝑟32+. … … … + 𝑟𝑛2)/𝑛]
𝐼 = 𝑀[(𝑟12 + 𝑟22 + 𝑟32+. … … … + 𝑟𝑛2)/𝑛]
𝑰 = 𝑴𝒌𝟐
k is called radius of gyration and is given by, 𝒌 = [(𝑟12 + 𝑟22 + 𝑟32+. … … … + 𝑟𝑛2)/𝑛]1/2
SI unit of radius of gyration is metre(m).
Theorems of perpendicular and parallel axis
These are two useful theorems, which are used to find the moment of inertia of regular shaped bodies about any axis of rotation if the value of the moment of inertia of the given body is known about a certain axis of rotation of the body.
Theorem of perpendicular axis
It states that the moment of inertia of a planar body about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.
Explanation: Let X, Y and Z are the mutually perpendicular axes. Consider an object in X-Y plane. Let 𝐼𝑧 → moment of inertia of the body about 𝑍 − 𝑎𝑥𝑖𝑠
𝐼𝑥 → moment of inertia of the body about X – axis
𝐼𝑦 → moment of Inertia of the body about Y-axis
Then 𝑰𝒛 = 𝑰𝒙 + 𝑰𝒚
Proof: Consider a particle of mass m at point P (x,y) at a distance r from O. The moment of inertia of the particle about Z-axis is 𝐼𝑧 = 𝑚𝑟2
The moment of inertia of the whole body with respect to z-axis is 𝐼𝑧 = ∑𝑚𝑖𝑟𝑖2 Similarly, the moment of inertia of the body with respect to x-axis is 𝐼𝑥 = ∑𝑚𝑖𝑥𝑖2 and with respect to 𝑦 − 𝑎𝑥𝑖𝑠 is 𝐼𝑦 = ∑𝑚𝑖𝑦𝑖2
Now 𝐼𝑧 = ∑𝑚𝑖𝑟𝑖2
𝐼𝑧 = ∑𝑚𝑖(𝑥𝑖2 + 𝑦𝑖2) ∵ 𝑟2 = 𝑥2 + 𝑦2
𝐼𝑧 = ∑𝑚𝑖𝑟𝑖2 +∑𝑚𝑖𝑦𝑖2
𝑰𝒛 = 𝑰𝒙 + 𝑰𝒚
Note: This theorem is applicable to the bodies which are planar or flat whose thickness is very small compared to their other dimensions.
Theorem of parallel axes
It states that the moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes.
Explanation: Let Z’ be the axis about which the moment of inertia of the body is to be calculated. Z is the axis passing through the centre of mass of the body and parallel to Z’. Let a be the distance between the two axes. Then,
𝑰𝒁 ′ = 𝑰𝒛 + 𝑴𝒂𝟐
Proof: Consider a particle at P of mass m whose distance from Z axis is y. Moment of inertia of the particle about Z axis is = my2 The moment of Inertia of the body about 𝑍 is 𝐼𝑧 = ∑𝑚𝑖𝑦𝑖2
The moment of inertia of the particle about Z’ is = 𝑚(𝑦𝑖 − 𝑎)2
The moment of inertia of the body about 𝑍’ is given by, 𝐼𝑧′ = ∑𝑚𝑖(𝑦𝑖 − 𝑎)2
𝐼𝑧 = ∑𝑚𝑖(𝑦𝑖2 + 𝑎2 − 2𝑦𝑖𝑎)
𝐼𝑧 = ∑𝑚𝑖𝑦𝑖2 + ∑𝑚𝑖𝑎2 −∑𝑚𝑖2𝑦𝑖𝑎
𝐼𝑧 = ∑𝑚𝑖𝑦𝑖2 + 𝑎2∑𝑚𝑖 − 2𝑎∑𝑚𝑖𝑦𝑖
The centre of mass is at the origin then, 𝑅𝑐𝑚 = 0 and 𝑅𝑐𝑚 = (1/𝑀)∑𝑚𝑖𝑦𝑖
0 = (1/𝑀)∑𝑚𝑖𝑦𝑖 Since 𝑀 ≠ 0, ∑𝑚𝑖𝑦𝑖 = 0
Substituting in the above equation, 𝐼𝑧′ = 𝐼𝑧 + 𝑀𝑎2 − 2𝑎(0)
𝑰𝒛′ = 𝑰𝒛 + 𝑴𝒂𝟐
Note: This theorem is applicable to a body of any shape.
Kinematics of Rotational motion about a fixed axis
The kinematical quantities in rotational motion, angular displacement (𝜃), angular velocity (𝜔) and anglar acceleration (𝛼) respectively correspond to kinematic quantities in linear motion displacement (x), velocity (v) and acceleration (a). They are,
𝜔 = 𝜔0 + 𝛼𝑡
𝜃 − 𝜃0 = 𝜔0𝑡 + (1/2)𝛼𝑡2
𝜔2 = 𝜔02 + 2𝛼(𝜃 − 𝜃0)
Dynamics of rotational motion about a fixed axis
Consider a cross–section of a rigid body rotating about a fixed axis, perpendicular the plane of the paper. Consider a particle at P in x-y plane which describes a circular path of radius r with centre at C. In time ∆t the particle moves to the position P’ due to application of force 𝐹. (i) Work done by the force is, 𝑑𝑊 = 𝐹 ∙ 𝑑𝑠
𝑑𝑊 = 𝐹𝑑𝑠 cos𝜙
𝑑𝑊 = 𝐹(𝑟𝑑𝜃) cos𝜙
𝑑𝑊 = (𝑟𝑑𝜃)𝐹 cos𝜙
Now 𝜙 + 𝛽 = 90°, because displacement vector is in the direction of tangent to the circular path.
By resolving component of 𝐹 with respect to r and with respect to displacement vector as shown in figure, we get, 𝐹 cos𝜙 = 𝐹 sin𝛽
∴ 𝑑𝑊 = (𝑟𝑑𝜃) 𝐹 sin𝛽
𝑑𝑊 = ( 𝑟𝐹 sin 𝛽)𝑑𝜃 𝑑
𝑊 = (𝑟 × 𝐹 )𝑑𝜃 (∵ 𝑟 × 𝐹 = 𝑟𝐹 sin 𝛽)
Total work done is, 𝒅𝑾 = 𝝉 𝒅𝜽
This equation is similar to the expression 𝑑𝑊 = 𝐹𝑑𝑠
(ii) Now dividing the equation 𝑑𝑊 = 𝜏 𝑑𝜃 throughout by dt.
𝑑𝑊/𝑑𝑡 = 𝜏 𝑑𝜃/𝑑𝑡
𝑷 = 𝝉𝝎
Where 𝑑𝑊/𝑑𝑡 = 𝑃 (rate of doing work = power)
(iii) The rate at which the work is done on the body is equal to rate at which kinetic energy changes.
𝑑𝑊/𝑑𝑡 = 𝑑/𝑑𝑡(𝐾𝐸)
𝑃 = 𝑑/𝑑𝑡(𝐼𝜔2/2)
𝜏𝜔 = 𝐼(2𝜔/2)(𝑑𝜔/𝑑𝑡)
𝜏𝜔 = 𝐼𝜔𝛼
𝝉 = 𝑰𝜶 which is similar to F = ma
Angular momentum in case of rotation about of fixed axis
Consider a rigid body of mass M rotating with an angular velocity 𝜔 along z-axis. The rigid body is made up of large number of elements. Consider one such element of mass mi whose position vector is 𝑟𝑖 and linear momentum is 𝑝𝑖. The angular momentum of this element about the axis of rotation is given by,
𝑙𝑖 = 𝑟𝑖 × 𝑝𝑖 = (𝑟𝑝 sin 90°)𝑘̂
𝑙𝑖 = (𝑟𝑖𝑝𝑖)𝑘̂
𝑙𝑖 = (𝑟𝑖𝑚𝑖𝑣𝑖)𝑘̂
𝑙𝑖 = (𝑟𝑖𝑚𝑖𝑟𝑖𝜔)𝑘̂
𝑙𝑖 = (𝑚𝑖𝑟𝑖2𝜔)𝑘̂
The total angular momentum is given by, 𝐿 = ∑(𝑚𝑖𝑟𝑖2𝜔)𝑘̂
𝐿 = (𝜔𝑘̂)∑𝑚𝑖𝑟𝑖2
𝐿 = (𝜔𝑘̂)𝐼
𝑳 = (𝑰𝝎)𝒌̂ or 𝑳 = 𝑰𝝎
Law of conservation of angular momentum
In the absence of external torque, the net angular momentum of the system is conserved.
Explanation: We have 𝐿 = 𝐼𝜔
Differentiating both sides with respect to t
𝑑𝐿/𝑑𝑡 = 𝑑/𝑑𝑡(𝐼𝜔)
But 𝑑𝐿/𝑑𝑡 = 𝜏𝑛𝑒𝑡 ∴ 𝑑/𝑑𝑡(𝐼𝜔) = 𝜏𝑛𝑒𝑡
If the net external torque acting on that body is zero then, 𝑑/𝑑𝑡(𝐼𝜔) = 0
𝑰𝝎 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
This is the law of conservation of angular momentum.
Note: If the M.I of the body changes from 𝐼1 to 𝐼2 due to the change of the distribution of mass of the body, then angular velocity of the body changes from 𝜔1 to 𝜔2 such that, 𝐼1𝜔1 = 𝐼2𝜔2 or 𝐼1𝜔1 = 𝐼2𝜔2
Illustrations for conservation of Angular momentum
1. Suppose a man is sitting on a rotating table with his arms stretched outward. When the man with draws his arms towards his chest, the moment of inertia of the man decreases. Hence his angular speed increases.
2. A ballet dancer varies her angular speed by stretching her legs and arms out or in. As she stretches her legs and arms out, her moment of inertia increases and angular speed decreases.
3. When a diver Jumps from the spring board he curls his body by rolling his arms and legs in, by doing so he decreases his moment of inertia and hence angular speed increases.
4. When a planet revolving around the sun in an elliptical orbit comes near the sun, its speed increases. This is because as the planet comes near the sun, its moment of inertia decreases and hence its angular velocity increases.
The combination of Rotational motion (without slipping) and the translational motion of a rigid body is known as rolling motion. Consider a spherical rigid body of radius R rolling over a horizontal surface. Let 𝜔 be the angular velocity of the rigid body about the axis of rotation. When this rigid body rolls over a smooth horizontal surface it has two types of motions simultaneously,
(i) The rotational motion about its centre of mass.
(ii) Translational motion of the centre of mass of the body.
The body will roll over the surface without slipping if the point of contact (say A) of the body with the surface is at rest at any contact with respect to the centre of mass. The magnitude of linear velocity, 𝑣𝑐𝑚 = 𝑅𝜔 at A. The velocity of the point B with respect to centre of mass = 𝜔𝑅 + 𝑣𝑐𝑚 = 2𝜔𝑅 or = 2𝑣𝑐𝑚
The velocity of the top most point on the body is maximum with respect to the centre of mass of the body.
Kinetic energy of rolling motion
The kinetic energy of a system of particles (K) can be separated into the kinetic energy of motion of the centre of mass (1/2)𝑚𝑣𝑐𝑚2 and kinetic energy of rotational motion about the centre of mass of the system of particles (1/2)𝐼𝜔2
𝐾 = (1/2)𝑚𝑣𝑐𝑚2 + (1/2)𝐼𝜔2
Where 𝑚 ⟶ mass of the body
𝑣𝑐𝑚 ⟶ velocity of centre of mass
𝐼 ⟶ moment of Inertia of the body
𝜔 ⟶ angular velocity of the body
𝐾 = (1/2)𝑚𝑣𝑐𝑚2 + (1/2)(𝑚𝑘2)(𝑣𝑐𝑚2𝑅2)
𝐾 = (1/2)𝑚𝑣𝑐𝑚2 + (1/2)𝑚𝑣𝑐𝑚2𝑘2𝑅2
𝐾 = (1/2)𝑚𝑣𝑐𝑚2(1 + 𝑘2𝑅2)
This equation applies to any rolling body: a disc, a cylinder a ring or a sphere.
1. Find the centre of mass of three particles at the vertices of an equilateral triangle. The masses of the particles are 100g, 150g and 200g respectively. Each side of the equilateral triangle is 0.5m long.
Co – ordinates of the mass m1 are, 𝑥1 = 0, 𝑦1 = 0.
Co-ordinates of the mass m2 are, 𝑥2 = 0.5, 𝑦2 = 0.
Co-ordinates of the mass m3 are, 𝑥3 = 0.5/2 = 0.25, 𝑦3 = (√3/2)(0.5) = 0.25√3
Let (𝑋𝑐𝑚, 𝑌𝑐𝑚) be the co-ordinates of the centre of mass
𝑋𝑐𝑚 = (𝑚1𝑥1 + 𝑚2𝑥2 + 𝑚3𝑥3)/(𝑚1 + 𝑚2 + 𝑚3) = (100(0) + 150(0.5) + 200(0.25))/(100 + 150 + 200𝑔) = (75 + 50)/450 = 125/450 = 5/18.
𝑌𝑐𝑚 = (𝑚1𝑦1 + 𝑚2𝑦2 + 𝑚3𝑦3)/(𝑚1 + 𝑚2 + 𝑚3) = [100(0) + 150(0) + 200(0.25√3)]/ (100 + 150 + 200𝑔) = (50√3)/450 = √3/9 = 1/3√3
(𝑋𝑐𝑚, 𝑌𝑐𝑚) = (5/18, 1/3√3 ) 𝑜𝑟 𝑅 = 5/18 𝑖+ 1/3√3𝑗
2. Find the centre of mass of a uniform L-shaped lamina with dimensions as shown. The mass of the lamina is 3kg.
Now we can divide the Lamina in to 3 squares of each length 1m and the mass of each lamina is 1kg, since it is uniform. Then C1, C2, C3 are the centre of masses of each square and the coordinates are 𝐶1(1/2, 3/2) , 𝐶2(1/2, 1/2) and 𝐶3(3/2, 1/2) because the geometric centres of the square is the centre of mass of the square. Now the centre of mass of lamina is the centre of mass of these three masses.
∴ 𝑋𝑐𝑚 = (𝑚1𝑥1 + 𝑚2𝑥2 + 𝑚3𝑥3)/(𝑚1 + 𝑚2 + 𝑚3) = [1(1/2) + 1(1/2) + 1(3/2)]/(1 + 1 + 1)= (5/2)/3 = 5/6
𝑌𝑐𝑚 = (𝑚1y1 + 𝑚2y2 + 𝑚3y3)/(𝑚1 + 𝑚2 + 𝑚3) = [1(3/2) + 1(1/2) + 1(1/2)]/(1 + 1 + 1) = 5/6
The co-ordinates of the centre of mass of L-shaped Lamina is,
(𝑋𝑐𝑚, 𝑌𝑐𝑚) = (5/6, 5/6)
Its position vector is 𝑅= 5/6𝑖 + 5/6𝑗
3. In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27Å. Find the approximate location of the CM of the molecule. Given that a chlorine atom is about 35.5 times as massive as hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.
Let these two atoms lay on the x-axis. Its centre of mass lie at (𝑋𝑐𝑚, 0)
𝑚1 = 𝑚𝐻 = 1 𝑢𝑛𝑖𝑡
𝑚2 = 𝑚𝐶𝑙 = 35.5 𝑢𝑛𝑖𝑡
∴ 𝑋𝑐𝑚 = (𝑚1𝑥1 + 𝑚2𝑥2)/(𝑚1 + 𝑚2) = [1(0) + 35.5 (1.27 × 10−10)]/(1 + 35.5) = (35.5/36.5) × 1.27 × 10−10 = 1.235Å
(𝑋𝑐𝑚, 0) = (1.235Å, 0)
𝑅 = 1.235 × 10−10𝑖
4. Find the scalar and vector product of two vector 𝐴 = 3𝑖− 4𝑗+ 5𝑘 and 𝐵 = −2𝑖+ 𝑗−3𝑘.
(i) 𝐴 ∙ 𝐵 = 𝐴𝑥𝐵𝑥 + 𝐴𝑦𝐵𝑦 + 𝐴𝑧𝐵𝑧 = [(3)(−2) + (−4)(1) + (5)(−3)] = −6 − 4 − 15 = −25
= 𝑖[(−4)(−3) − 5] − 𝑗[(3)(−3) − (−2)(5)] + 𝑘[(3)(1) − (−2)(−4)] = 𝑖[12 − 5] − 𝑗[−9 + 10] + 𝑘[3 − 8] = 7𝑖− 1𝑗− 5𝑘
5. Show that 𝐴 = −6𝑖+ 9𝑗− 12𝑘 and 𝐵̂ = 2𝑖− 3𝑗+ 4𝑘 are parallel to each other. 𝐴 and 𝐵 will be parallel to each other if 𝐴 × 𝐵 = 0.
𝐴 × 𝐵 = 𝑖[(9)(4) − (−12)(−3)] − 𝑗[(−6)(4) − (−12)(2)] + 𝑘[(−6)(−3) − (9)(2)]
𝐴 × 𝐵 = 𝑖[36 − 36] − 𝑗[−24 + 24] + 𝑘[+18 − 18]
𝐴 × 𝐵 = 0
Since 𝐴 × 𝐵 = 0 therefore 𝐴 and 𝐵 are parallel to each other.
6. Magnitude of the cross product of two vectors 𝐴 and 𝐵 represents the area of the parallelogram. Prove it.
Consider a parallelogram OPQR whose adjacent sides are QP and OR are represented both in magnitude and direction by two vectors 𝐴 and 𝐵
|𝐴 × 𝐵| = 𝐴𝐵 sin 𝜃 = 𝐴(𝐵 sin 𝜃) = 𝑂𝑃. 𝑅𝑁 = 𝐵𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡
|𝐴 × 𝐵| = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑡𝑜𝑔𝑟𝑎𝑚
Note: Since the parallelogram has two triangles OPQ and ORQ of equal areas,
Area of each triangle = (1/2)|𝐴 × 𝐵|
Thus area of a triangle contained between the vectors 𝐴 and 𝐵 is one half of the magnitude of 𝐴 and 𝐵 is one half of the magnitude of 𝐴 × 𝐵.
7. Find the torque of a force 7𝑖+ 3𝑗− 5𝑘 about the origin. The force acts on a particle whose position vector is 𝑖− 𝑗+ 𝑘. Given 𝑟 = 𝑖− 𝑗+ 𝑘 and 𝐹 = 7𝑖+ 3𝑗− 5𝑘 Torque 𝜏 = 𝑟 × F
= 𝑖[(1 − 1)(−5) − (1)(3)] − 𝑗[(1)(−5) − (1)(7)] + 𝑘[(1)(3) − (−1)(7)] = 𝑖[+5 − 3] − 𝑗[−5 − 7] + 𝑘[3 + 7]
𝜏 = 2𝑖+ 12𝑗+ 10𝑘
8. A meter scale is balanced on a knife edge at its centre. When two coins each of mass 5g are put one on top of the other at the 12.0 cm mark, the stick is found to be balanced at 45.0 cm what is the mass of the meter stick?
From the principle of moments, 𝐹1𝑑1 = 𝐹2𝑑2
𝑚𝑔 × (45 − 12) = 𝑀𝑔 × (5)
𝑚𝑔 × 33 = 𝑀𝑔 × 5
10 × 𝑔 × 33 = 𝑀 × 𝑔 × 5
𝑀 = (10 ×33)/5 = 66 𝑔𝑟𝑎𝑚
9. A car weights 1800 kg the distance between its front and back axel is 1.8 m its centre of gravity is 1.05 m behind the front axle. Determine the force exerted by the level road on each front wheel and each back wheel.
Let F1, F2 and Mg be the force exerted by the ground on front wheels, back wheels and weight of the car respectively. Now car is at rest. For translational equilibrium,
𝐹1 + 𝐹2 − 𝑀𝑔 = 0
𝐹1 + 𝐹2 = 𝑀𝑔
and also car is in rotational equilibrium
Torque on front wheels = 0
(𝐹2 × 1.8) – (𝑀𝑔 × 1.05) = 0
𝐹2 = (𝑀𝑔×1.05)/1.8 = (1800×9.8×1.05)/1.8
𝐹2 = 10290𝑁
Force on each back wheel = 5145N
Torque on rare wheels = 0
(𝐹1 × 1.8) – (𝑀𝑔 × 0.75) = 0
𝐹1 = (𝑀𝑔×1.07)/1.8 = (1800×9.8×0.75)/1.8
𝐹2 = 7350N
Force on each Front wheel = 3675 N
10. A non-uniform bar of weight w is suspended at rest by two strings of negligible weight as shown the angles made by the strings with the vertical re 26.90 and 53.10 respectively. The bar is 2m long calculate the distance d of the centre of gravity of the bar from its left end.
Since the bar is at rest so the net external force and net external torque on the bar is zero.
∑𝐹𝑥 = 0 and ∑𝐹𝑦 = 0
∑𝐹𝑥 = 𝑇1 sin 𝜃 − 𝑇2 sin 𝜃 = 0
𝑇1 sin(36.9) − 𝑇2 sin(53.1) = 0
(𝑇1 × 0.6004) – (𝑇2 × 0.7996) = 0
𝑇2 × (0.6004/0.7996)𝑇1 = 0.7508 𝑇1 − − − −> (1)
∑𝐹𝑦 = 𝑇1 cos 𝜃 + 𝑇2 cos 𝜃 − 𝑀𝑔 = 0
(𝑇1 cos(36.9)) + (𝑇2 × cos(53.1)) − 𝑀𝑔 = 0
(𝑇1 × 0.7996) + (𝑇2 × 0.6004) − 𝑀𝑔 = 0
0.7996𝑇1 + 0.6004𝑇2 = 𝑀𝑔 ——-à (2)
Now torque acting on the left end of the bar is = 0
(𝑇2 cos 𝜃 × 𝑙) − 𝑀𝑔𝑑 = 0
(𝑇2 cos(53.1) × 2) – (𝑀𝑔 × 𝑑) = 0
(𝑇2 × 0.6004 × 2) − 𝑀𝑔𝑑 = 0
1.2008𝑇2 = 𝑀𝑔𝑑 ——-à (3)
1.2008𝑇2 = (0.7996𝑇1 + 0.6004𝑇2)𝑑 from (2)
1.2008 (0.7508𝑇1) = [0.7996𝑇1 + 0.6004(0.7508)𝑇1]𝑑 from (1)
0.9015𝑇1 = (0.7996𝑇1 + 0.4507𝑇1)𝑑
0.9015𝑇1 = 1.2503𝑇1𝑑
𝑑 = 0.9015𝑇1/1.2503𝑇1 = 0.72 𝑚
11. What is the moment of a disc about one of its diameters? We have, moment of Inertia of inertia of the disc about an axis perpendicular to it is given by, 𝐼𝑧 = 𝑀𝑅22
Where 𝑀 → mass of the disc
𝑅 → Radius of the disc
According to the perpendicular axes theorem, 𝐼𝑧 = 𝐼𝑥 + 𝐼𝑦
X and Y axes are along two diameters, ∴ 𝐼𝑥 = 𝐼𝑦
𝐼𝑧 = 𝐼𝑥 + 𝐼𝑥
𝐼𝑧 = 2𝐼𝑥
But 𝐼𝑧 = 𝑀𝑅2/2
𝑀𝑅2/2 = 2𝐼𝑥
𝐼𝑥 = 𝑀𝑅2/4
12. Find the moment of inertia of a sphere about a tangent to the sphere, given the moment of inertia of a sphere about any of its diameter to be 2𝑚𝑅2/5, where m is the mass of the sphere and R is the radius of the sphere.
Let AB is a diameter of the given sphere. M.I of the sphere about its diameter 𝐴𝐵 = 𝐼𝐴𝐵 = (2/5)𝑚𝑅2
According to the theorem of parallel axis
𝐼𝐶𝐷 = 𝐼𝐴𝐵 + 𝑚𝑅2 = (2/5)𝑚𝑅2 + 𝑚𝑅2
𝐼𝐶𝐷 = 𝑚𝑅2((2/5) + 1)
𝐼𝐶𝐷 = (7/5)𝑚𝑅2
13. What is the moment of inertia of a rod of mass M, length l about an axis perpendicular to it through one end?
The moment of inertia of the body about an axis passing through its centre is given by,
𝐼𝐴𝐵 = 𝑀𝑙2/12
Using the parallel axis theorem,
𝐼𝐶𝐷 = 𝐼𝐴𝐵 + 𝑀(𝑙/2)2
𝐼𝐶𝐷 = (𝑀𝑙2/12) + (𝑀𝑙2/4) = 𝑀𝑙2[(1/12) + (1/4)]
𝐼𝐶𝐷 = 𝑀𝑙2[4/12] = 𝑀𝑙2/3
14. What is the moment of inertia of a ring about a tangent to the circle of the ring? The tangent to the ring in the plane of the ring is parallel to the diameter of the ring.
According to parallel axis theorem, 𝐼𝑡𝑎𝑛 = 𝐼𝑑𝑖𝑎 + 𝑀𝑅2
Where 𝑀 → mass of the ring
𝑅 → Radius of the ring
𝐼𝑡𝑎𝑛 = 𝑀𝑅2/2 + 𝑀𝑅2
𝐼𝑡𝑎𝑛 = (3/2)𝑀𝑅2
15 Given the momentum of inertia of disc of mass m and radius R about any of its diameter to be 𝑚𝑅2/4. Find its moment of inertia about an axis normal to the disc and passing through a point on its edge.
M. I about any diameter = (1/4)𝑚𝑅2
𝐼𝑥 = 𝐼𝑦 = (1/4)𝑚𝑅2
By perpendicular axis theorem,
𝐼𝑧 = 𝐼𝑥 + 𝐼𝑦
𝐼𝑧 = (1/4)𝑚𝑅2 + (1/4)𝑚𝑅2
𝐼𝑧 = (1/2)𝑚𝑅2
Now Z and Z1 are parallel, according to parallel axis theorem,
𝐼𝑧1 = 𝐼𝑧 + 𝑚𝑅2
𝐼𝑧1 = (1/2)𝑚𝑅2 + 𝑚𝑅2
𝐼𝑧1 = (3/2)𝑚𝑅2
16. A solid cylinder of mass 20kg rotates about its axis with angular speed 100 rad/s the radius of the cylinder is 0.25 m. What is the kinetic energy associated with the rotation of the cylinder? What is the magnitude of angular momentum of the cylinder about its axis?
m = 20 kg, R = 0.25 m, 𝜔 = 100 rad/s, I=?, K = ?, L=?
M.I of the solid cylinder about its axis of symmetry,
𝐼 = (1/2)𝑚𝑅2
𝐼 = (1/2) × 0.625 × 100 × 100
𝐼 = 0.625kgm2
𝐾 = (1/2)𝐼𝜔2
𝐾 = (1/2) × 20 × 0.25 × 0.25 𝐾 = 3125𝐽
𝐿 = 𝐼𝜔
𝐿 = 0.625 × 100 = 62.5 𝐽𝑠
17. A child stands at the centre of a turn table with his two arms out stretched. The turn table is set rotating with an angular speed of 4 over / min how much is the angular speed of the child if he folds his hands back and there by reduces his moment of inertia to 2/5 times the initial value? Assume that the turn table rotates without friction. Show that the child’s new kinetic energy of rotation is more than the initial kinetic energy of rotation how do you account for this increase in kinetic energy?
According to the law of conservation of angular momentum,
𝐼1𝜔1 = 𝐼2𝜔2
𝜔2 = 𝐼1𝜔1/𝐼2
𝜔2 = (𝐼1 × 40)/((2/5)𝐼1) = (5 × 40)/2 = 100 𝑟𝑒𝑣/𝑚𝑖𝑛
𝑘𝑖/𝑘𝑓 = (1/2)𝐼1𝜔12/(1/2)𝐼2𝜔22
𝑘𝑖/𝑘𝑓 = [𝐼1 × (40)2]/[(2/5)𝐼1 × (100)2]
𝑘𝑖/𝑘𝑓 = (5/2) × (1600/10000) = (5/2) × (4/25)
𝑘𝑖/𝑘𝑓 = 2/5
𝐾𝑖 = (2/5)𝑘𝑓 𝑜𝑟 𝐾𝑓 = (5/2)𝑘𝑖
This increase in kinetic energy is obtained from the muscular energy of the child when he folds back his arms.
18. A rope of negligible mass is wound round a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30N? What is the linear acceleration of the rope? Assume that there is no slipping.
m = 3kg, R =40cm = 0.4m, F = 30N
M.I of the hollow cylinder
𝐼 = 𝑚𝑅2
𝐼 = 3 × 0.4 × 0.4 = 0.48𝑘𝑔𝑚2
𝑇𝑜𝑟𝑞𝑢𝑒, 𝜏 = 𝑟 × 𝐹
𝜏 = 0.4 × 30 = 12𝑁𝑚
𝛼 = 𝜏/𝐼 = 12/0.48 = 25𝑟𝑎𝑑/𝑠2
We have 𝑣 = 𝑟𝜔
𝑑𝑣/𝑑𝑡 = 𝑑/𝑑𝑡(𝑟𝜔) = 𝑟(𝑑𝜔/𝑑𝑡)
𝑎 = 𝑟𝛼 = 0.4 × 25 = 10𝑚𝑠−2
19. To maintain a rotor at a uniform angular speed of 200rad/s, an engine needs to transmit a torque of 180Nm. What is the power required by the engine? Assume that the engine is 100% efficient.
𝜔 = 200 𝑟𝑎𝑑/𝑠 Power required, P = 𝜏𝜔, 𝜏 = 180𝑁𝑚
𝑃 = 180 × 200
𝑃 = 36000𝑤
𝑃 = 36 𝑘𝑊
20. A hoop of radius 2m, weights 100kg. It rolls along a horizontal floor so that its centre of mass has a speed of 20cm/s. How much work has to be done to stop it?
Work done to stop the hoop = Kinetic energy of the hoop.
𝑊 = (1/2)𝑚𝑣𝑐𝑚2 + (1/2)𝐼𝜔2
𝑊 = (1/2)𝑚𝑣𝑐𝑚2 + (1/2)(𝑚𝑅2)(𝑣𝑐𝑚/𝑅)2
𝑊 = (1/2)𝑚𝑣𝑐𝑚2 + (1/2)𝑚𝑣𝑐𝑚2 = 𝑚𝑣𝑐𝑚2
𝑊 = 100 × 0.2 × 0.2 = 4 𝐽
21. The oxygen molecule has a mass of 5.36 × 10−26𝑘𝑔 and a moment of inertia of 1.94 × 10−46𝑘𝑔𝑚2 about an axis through its centre perpendicular to the lines joining the two atoms. Suppose the mean speed of such a molecule in a gas is 500m/s and that is K.E of rotation is 2/3 of its K.E of translation find the average angular velocity of the molecule.
𝑚 = 5.36 × 10−26𝑘𝑔
𝐼 = 1.94 × 10−46𝑘𝑔/𝑚2
𝑣 = 500𝑚/𝑠
Given, K.E of rotation = (2/3) × 𝐾. 𝐸 of translation
(1/2)𝐼𝜔2 = (2/3) × (1/2)𝑚𝑣2
𝜔2 = 𝑚𝑣2/3𝐼
𝜔 = √[(2 × 5.30 × 10−26 × 500 × 500)/(3 × 1.94 × 10−46)] = 6.7478 × 102 𝑟𝑎𝑑/s
Important Questions for Exam
1. What is a rigid body?
2. What is the angle between 𝑃 × 𝑄 and 𝑄 × 𝑃?
3. Is 𝑃 × 𝑄 = 𝑄 × 𝑃?
4. Define angular displacement.
5. Mention the SI unit of angular velocity.
6. Define Torque.
7. Write the dimensional formula for torque.
8. Define angular momentum of a rigid body.
9. State the law of conservation of angular momentum.
1. Define Angular velocity and angular acceleration.
2. Give two general conditions of equilibrium of a rigid body. or Write the conditions for equilibrium of a rigid body.
3. What is couple? Mention effect of couple acting on a body.
4. Define moment of inertia and mention its SI unit.
5. Mention the formula for moment of inertia of a solid cylinder about its axis and explain the symbols.
6. Define radius of gyration. Is it constant for a body?
1. What is the value of (a) 𝑖 × 𝑖 (b) 𝑗 × 𝑗 (c) 𝑘 × 𝑘
2. What is the value of (a) 𝑖 × 𝑗 (b) 𝑗 × 𝑘 (c) 𝑘 × 𝑖
3. What is the value of (a) 𝑗 × 𝑖 (b) 𝑘 × 𝑗 (c) 𝑖 × 𝑘
4. What is centre of mass? Write the expression for its position in a two body system using a diagram and explain each term.
5. Distinguish between scalar product and vector product.
6. Define vector product of two vectors. Write an expression for it explaining each them.
7. Derive the relation between linear velocity and angular velocity.
8. Explain the principle of lever.
9. Write the expression for moment of inertia of a circular disc about an axis perpendicular to it at the centre and give the expression for its radius of gyration.
10. Compare the translation motion and rotational motion. or Compare the equations of linear and rotational motions.
1. State and explain theorems of perpendicular and parallel axis. or State and explain theorems of moment of inertia.
2. Define torque. Show that the rate of change of angular momentum is equal to torque acting on the system. or Deduce an expression for torque, 𝜏 = 𝑑𝑙/𝑑𝑡 𝐨𝐫 Derive the relation 𝑑𝑙/𝑑𝑡 = 𝜏 with usual notation or Derive the relation between torque and angular momentum.
3. Define torque. State and explain perpendicular and parallel axis theorem.
4. Derive an expression for Kinetic energy of a rolling body.