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PHYSICS CLASS XII Chapter 1 - Rotationall dynamics

1. The document discusses rotational dynamics and circular motion. It defines concepts like angular velocity, moment of inertia, centripetal force, and radius of gyration. 2. Examples of circular motion discussed include vehicles moving in circular tracks, wells of death, and vehicles on banked roads. The forces and equations of motion are analyzed. 3. Vertical circular motion under gravity is also examined, like a point mass attached to a string or rod moving in a vertical circle. Dynamics of a vehicle on a convex overbridge are also covered. 4. Moment of inertia is introduced as an analogous concept to mass for rotational motion. Formulas are given for moment of inertia of objects like rings, discs, and

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1 PHYSICS CLASS XII CHAPTER I ------------------------ ROTATIONAL DYNAMICS 2021-22
2 1.1 INTRODUCTION Every day we come across several revolving or rotating (rigid) objects. During revolution, the object undergoes circular motion about some point outside the object or about some other object, while during rotation the motion is about an axis of rotation passing through the object. 1.2 Characteristics of Circular Motion: 1) It is an accelerated motion 2) It is a periodic motion 1.2.1 Kinematics of Circular Motion: The tangential velocity is, 𝒗 ⃗ ⃗ = 𝒘 ⃗⃗⃗ x 𝒓 ⃗ , where 𝒘 ⃗⃗⃗ is the angular velocity. The magnitude of 𝒗 ⃗ ⃗ is v = ω r. Direction of 𝒘 ⃗⃗⃗ is always along the axis of rotation and is given by the right-hand thumb rule. Fig. Directions of angular velocity If T is period of circular motion and n is the frequency, w = 2𝝅n = 𝟐𝝅 𝑻 Uniform circular motion: During circular motion if the speed of the particle remains constant, it is called Uniform Circular Motion (UCM). The acceleration responsible for this is the centripetal or radial acceleration 𝒂𝒓 ⃗⃗⃗⃗ = - 𝒘𝟐 𝒓 ⃗ For UCM, its magnitude is constant and it is a = 𝒘𝟐 r = 𝒗𝟐 𝒓 = vw. It is always directed towards the centre of the circular motion, hence called centripetal. Fig. Directions of linear velocity and acceleration. Non-uniform circular motion: For non-uniform circular motion, the magnitude of 𝒂𝒓 ⃗⃗⃗⃗ is not constant. This is due to the angular acceleration 𝜶 ⃗⃗ = 𝒅𝒘 ⃗⃗ ⃗ 𝒅𝒕 For increasing speed, it is along the direction of 𝒘 ⃗⃗⃗ while during decreasing speed, it is opposite to that of 𝒘 ⃗⃗⃗ . Fig. Direction of angular acceleration. If the angular acceleration 𝜶 ⃗⃗ is constant and along the axis of rotation, all 𝜽 ⃗ ⃗ , 𝒘 ⃗⃗⃗ and 𝜶 ⃗⃗ will be directed along the axis. 1.2.2 Dynamics of Circular Motion (Centripetal Force and Centrifugal Force):
3 i) Centripetal force (CPF): The acceleration responsible for circular motion is the centripetal or radial acceleration 𝒂𝒓 ⃗⃗⃗⃗ = - 𝒘𝟐 𝒓 ⃗ . The force providing this acceleration is the centripetal or radial force, CPF = −𝒎𝒘𝟐 𝒓 ⃗ ii) Centrifugal force (c.f.f.): This force, (+𝒎𝒘𝟐 𝒓 ⃗ ) the centrifugal force. It is a pseudo force arising due to the centripetal acceleration of the frame of reference. It must be understood that centrifugal force is a non-real force, but NOT an imaginary force. Resultant force = −𝒎𝒘𝟐 𝒓 ⃗ Or 𝒎𝒘𝟐 𝒓 ⃗ + ∑(𝒓𝒆𝒂𝒍 𝒇𝒐𝒓𝒄𝒆𝒔) = 0 1.3 Applications of Uniform Circular Motion: 1.3.1 Vehicle Along a Horizontal Circular Track: Forces acting on the car (considered to be a particle) are (i) weight mg, vertically downwards, (ii) normal reaction N, vertically upwards that balances the weight mg and (iii) force of static friction 𝒇𝒔 between road and the tyres. This is static friction because it prevents the vehicle from outward slipping or skidding. This is the resultant force which is centripetal. Fig. Vehicle on a horizontal road ∴ 𝒎𝒈 = 𝑵 𝒂𝒏𝒅 𝒇𝒔 = 𝒎𝒓𝒘𝟐 = 𝒎𝒗𝟐 𝒓 ∴ 𝒇𝒔 𝑵 = 𝒓𝒘𝟐 𝒈 = 𝒗𝟐 𝒓𝒈 For a given track, radius r is constant. For given vehicle, mg = N is constant. (𝒇𝒔)𝒎𝒂𝒙 = 𝝁𝒔. 𝑵, where, 𝝁𝒔 is the coefficient of static friction At the maximum possible speed 𝒗𝒔, (𝒇𝒔)𝒎𝒂𝒙 𝑵 = 𝝁𝒔 = 𝒗𝒎𝒂𝒙 𝟐 𝒓𝒈 ∴ 𝒗𝒎𝒂𝒙 = √𝝁𝒔𝒓𝒈 1.3.2 Well (or Wall) of Death: (मौत का क ु आँ): The forces acting on the vehicle are (i) Normal reaction N acting horizontally and towards the centre, (ii) Weight mg acting vertically downwards, and (iii) Force of static friction 𝒇𝒔 acting vertically upwards between vertical wall and the tyres. Fig. Well of death N = mr𝒘𝟐 = 𝒎𝒗𝟐 𝒓 and mg = 𝒇𝒔 Force of static friction 𝒇𝒔 is always less than or equal to 𝝁𝒔N. ∴ 𝒇𝒔 ≤ 𝝁𝒔 ( 𝒎𝒗𝟐 𝒓 )
4 ∴ 𝒈 ≤ 𝝁𝒔𝒗𝟐 𝒓 ∴ 𝒗𝟐 ≥ 𝒓𝒈 𝝁𝒔 ∴ 𝑽𝒎𝒊𝒏 = √ 𝒓𝒈 𝝁𝒔 1.3.3 Vehicle on a Banked Road: “Defined as the phenomenon in which the edge is raised for the curved roads above the inner edge to provide the necessary centripetal force to the vehicle to that they take a safe turn” There are two forces acting on the vehicle, (i) weight mg, vertically downwards and (ii) normal reaction N, perpendicular to the surface of the road. Its vertical component Ncosθ balances weight mg. Horizontal component Nsinθ being the resultant force, must be the necessary centripetal force. Fig: Vehicle on a banked road. N cos 𝜽 = mg and N sin 𝜽 = mr𝒘𝟐 = 𝒎𝒗𝟐 𝒓 ∴ 𝒕𝒂𝒏 𝜽 = 𝒗𝟐 𝒓𝒈 (a) Most safe speed: For a particular road, r and θ are fixed. 𝒗𝒔 = √𝒓𝒈 𝒕𝒂𝒏 𝜽 (b) Banking angle: While designing a road, this expression helps us in knowing the angle of banking as 𝜽 = 𝒕𝒂𝒏−𝟏 ( 𝒗𝟐 𝒓𝒈 ) (c) Speed limits: If the vehicle is running exactly at the speed 𝒗𝒔 = √𝒓𝒈 𝒕𝒂𝒏 𝜽, the forces acting on the vehicle are (i) weight mg acting vertically downwards and (ii) normal reaction N acting perpendicular to the road. Fig: Banked road: lower speed limit For speeds 𝒗𝟏 < √𝒓𝒈 𝒕𝒂𝒏 𝜽, 𝒎𝒗𝟏 𝟐 𝒓 < 𝑵 𝒔𝒊𝒏 𝜽.
5 Fig: Banked road: upper speed limit. These two forces take care of the necessary centripetal force. ∴ 𝒎𝒈 = 𝒇𝒔 𝒔𝒊𝒏 𝜽 + 𝑵 𝒄𝒐𝒔 𝜽 and For minimum possible speed, 𝒇𝒔 is maximum and equal to 𝝁𝒔N. For 𝝁𝒔 ≥ tan 𝜽, 𝒗𝒎𝒊𝒏 = 0. This is true for most of the rough roads, banked at smaller angles. (d) For speeds 𝒗𝟐 = √𝒓𝒈 𝒕𝒂𝒏 𝜽, 𝒎𝒗𝟐 𝟐 𝒓 > 𝑵 𝒔𝒊𝒏 𝜽. For maximum possible speed, 𝒇𝒔 is maximum and equal to 𝝁𝒔N. If 𝝁𝒔 = cot θ, 𝒗𝒎𝒂𝒙 = ∞. But (𝝁𝒔)𝒎𝒂𝒙 = 1. Thus, for θ ≥ 45°, 𝒗𝒎𝒂𝒙 = ∞. (e) For 𝝁𝒔 = 0, both the equations give us v = √𝒓𝒈 𝒕𝒂𝒏 𝜽 which is the safest speed 1.3.4 Conical Pendulum: A tiny mass connected to a long, flexible, massless, inextensible string, and suspended to a rigid support is called a pendulum. The string moves along the surface of a right circular cone of vertical axis and the point object performs a (practically) uniform horizontal circular motion is called a conical pendulum. Fig: In an inertial frame The forces acting on the bob are (i) its weight mg directed vertically downwards and (ii) the force 𝑻𝟎 due to the tension in the string, directed along the string, towards the support A.
6 Fig: In a non- inertial frame Radius r of the circular motion is r = L sin 𝜽. 1.4 Vertical Circular Motion: Two types of vertical circular motions are: (a) A controlled vertical circular motion such as a giant wheel or similar games. In this case the speed is either kept constant or NOT totally controlled by gravity. (b) Vertical circular motion controlled only by gravity. In this case, we initially supply the necessary energy (mostly) at the lowest point. 1.4.1 Point Mass Undergoing Vertical Circular Motion Under Gravity: Case I: Mass tied to a string: Both, weight (mg) and force due to tension are downward i.e., towards the center. There are only resultant centripetal force acts towards centre. Fig: Vertical circular motion Uppermost position (A): Thus, if 𝑽𝑨 is the speed at the uppermost point, Radius r of the circular motion is the length of the string. For minimum possible speed, Lowermost position (B): Force due to the tension, 𝑻𝑩 is vertically upwards, and opposite to mg. If 𝑽𝑩 is the speed at the lowermost point, The vertical displacement is 2r and the motion is governed only by gravity.
7 Positions when the string is horizontal (C and D): Force due to the tension is the only force towards the centre as weight mg is perpendicular to the tension. Arbitrary positions: Force due to the tension and weight are neither along the same line, nor perpendicular. It decreases the speed while going up and increases it while coming down. Case II: Mass tied to a rod: Consider a bob tied to a rod and whirled along a vertical circle. Zero speed is possible at the uppermost point. 1.4.2 Sphere of Death (मृत्यु गोल): Two-wheeler rider undergo rounds inside a hollow sphere. Starting with small horizontal circles, they eventually perform revolutions along vertical circles. The dynamics of this vertical circular motion is the same as that of the point mass tied to the string, except that the force due to tension T is replaced by the normal reaction force N. The linear speed is more for larger circles but angular speed is more for smaller circles. This is as per the theory of conical pendulum. 1.4.3 Vehicle at the Top of a Convex Over-Bridge: Fig: Vehicle on a convex over-bridge Forces acting on the vehicle are (a) Weight mg and (b) Normal reaction force N, both along the vertical line. Thus, if v is the speed at the uppermost point, As the speed is increased, N goes on decreasing. Thus, for just maintaining contact, N = 0. This imposes an upper limit on the speed as 𝒗𝒎𝒂𝒙 = √𝒓𝒈 1.5 Moment of Inertia as an Analogous Quantity for Mass: Fig: A body of N particles Let us consider the object to be consisting of N particles of masses 𝒎𝟏, 𝒎𝟐 .………𝒎𝒏 at respective perpendicular distances 𝒓𝟏,𝒓𝟐,… … … 𝒓𝒏 from the axis of rotation. Translational K.E. of the first particle is
8 Thus, rotational K.E. If I = ∑ 𝒎𝒊𝒓𝒊 𝟐 replaces mass m and angular speed ω replaces linear speed v, rotational K.E. = 𝟏 𝟐 𝑰𝒘𝟐 is analogous to translational K.E. = 𝟏 𝟐 𝒎𝒗𝟐 Thus, I is defined to be the rotational inertia or moment of inertia (M.I.) The moment of inertia of an object depends upon (i) individual masses and (ii) the distribution of these masses about the given axis of rotation. The moment of inertia is to be obtained by integration as I = ∫ 𝒓𝟐 𝒅𝒎 . Fig: Moment of Inertia of a ring 1.5.1 Moment of Inertia of a Uniform Ring: An object is called a uniform ring if its mass is situated uniformly on the circumference of a circle. If it is rotating about its own axis, its entire mass M is practically at a distance equal to its radius R form the axis. The moment of inertia of a uniform ring of mass M and radius R is I = 𝑴𝑹𝟐 . 1.5.2 Moment of Inertia of a Uniform Disc: The ratio 𝝈 = 𝒎 𝑨 = 𝒎𝒂𝒔𝒔 𝒂𝒓𝒆𝒂 is called the surface density. Consider a uniform disc of mass M and radius R rotating about its own axis, which is the line perpendicular to its plane and passing through its centre 𝝈 = 𝑴 𝝅𝑹𝟐. Fig: Moment of Inertia of a disk Width of this ring is dr, which is so small that the entire ring can be considered to be of average radius r. Area of this ring is A = 2πr.dr ∴ 𝝈 = 𝒅𝒎 𝟐𝝅𝒓.𝒅𝒓 ∴ dm = 2πσr.dr. The moment of inertia of this ring is 𝑰𝒓 = 𝒅𝒎 (𝒓𝟐 ) Moment of inertia (I) of the disc can now be obtained by integrating 𝑰𝒓 from r = 0 to r = R.
9 1.6 Radius of Gyration: It depends upon mass of that object and how that mass is distributed around the axis of rotation. The moment of inertia of any object as I = 𝑴𝑲𝟐 , where M is mass of that object. K is defined as the radius of gyration of the object about the given axis of rotation, I = 𝑴𝑲𝟐 . Larger the value of K, farther is the mass from the axis. 1.7 Theorem of Parallel Axes and Theorem of Perpendicular Axes: Expressions of moment of inertias of regular geometrical shapes are about their axes of symmetry. 1.7.1 Theorem of Parallel Axes: Fig: Theorem of parallel axes Consider, MOP is any axis passing through point O. Axis ACB is passing through the centre of mass C of the object, parallel to the axis MOP, and at a distance h from it (∴ h = CO). Consider a mass element dm located at point D. Perpendicular on OC from point D is DN. Moment of inertia of the object about the axis ACB is 𝑰𝑪 = ∫(𝑫𝑪)𝟐 𝒅𝒎, and about the axis MOP it is 𝑰𝟎 = ∫(𝑫𝟎)𝟐 𝒅𝒎 . From the definition of the centre of mass, ∫ 𝑵𝑪. 𝒅𝒎 = 𝟎 . It states that, “The moment of inertia (𝑰𝟎) of an object about any axis is the sum of its moment of inertia (𝑰𝑪) about an axis parallel to the given axis, and passing through the centre of mass and the product of the mass of the object and the square of the distance between the two axes (M𝒉𝟐 ).” 1.7.2 Theorem of Perpendicular Axes: Consider, 𝑰𝒙 and 𝑰𝒚 be the moment of Inertia of axis passing through point ‘M’ and ‘N’. Fig: Theorem of perpendicular axes Consider a mass element dm located at any point P. PM = y and PN = x are the perpendiculars drown from P respectively on the x and y axes.
10 z = √𝒚𝟐 + 𝒙𝟐. If 𝑰𝒙, 𝑰𝒚 and 𝑰𝒛 are the respective moment of inertias It states that, “The moment of inertia (𝑰𝒛) of a laminar object about an axis (z) perpendicular to its plane is the sum of its moment of inertias about two mutually perpendicular axes (x and y) in its plane, all the three axes being concurrent”. 1.8 Angular Momentum or Moment of Linear Momentum: 𝑳 ⃗ ⃗ = 𝒓 ⃗ × 𝒑 ⃗ ⃗ , were 𝒓 ⃗ is the position vector from the axis of rotation. In magnitude, L = P × r sin θ, where θ is the smaller angle between the directions of 𝑷 ⃗⃗ and 𝒓 ⃗ . 1.8.1 Expression for Angular Momentum in Terms of Moment of Inertia: A rigid object rotating with a constant angular speed ω about an axis perpendicular to the plane of paper. Let us consider the object to be consisting of N number of particles of masses 𝒎𝟏, 𝒎𝟐,… … … . 𝒎𝑵 at respective perpendicular distances 𝒓𝟏, 𝒓𝟐,… … … . 𝒓𝑵 from the axis of rotation. As the object rotates, all these particles perform UCM with same angular speed ω, but with different linear speeds 𝒗𝟏 = 𝒓𝟏𝒘, 𝒗𝟐 = 𝒓𝟐𝒘,… … … . 𝒗𝑵 = 𝒓𝑵𝒘. Directions of individual velocities 𝒗𝟏 ⃗⃗⃗⃗ , 𝒗𝟐 ⃗⃗⃗⃗ , etc., are along the tangents to their respective tracks. Linear momentum of the first particle is of magnitude 𝒑𝟏 = 𝒎𝟏𝒗𝟏 = 𝒎𝟏𝒓𝟏𝒘. Its direction is along that of 𝒗𝟏 ⃗⃗⃗⃗ . Its angular momentum is thus of magnitude 𝑳𝟏 = 𝒑𝟏𝒓𝟏 = 𝒎𝟏𝒓𝟏 𝟐 𝒘 Similarly, 𝑳𝟐 = 𝒎𝟐𝒓𝟐 𝟐 𝒘, 𝑳𝟑 = 𝒎𝟑𝒓𝟑 𝟐 𝒘, ………., 𝑳𝑵 = 𝒎𝑵𝒓𝑵 𝟐 𝒘 Thus, magnitude of angular momentum of the body is, Where, I = 𝒎𝟏𝒓𝟏 𝟐 + 𝒎𝟐𝒓𝟐 𝟐 + …. + 𝒎𝑵𝒓𝑵 𝟐 is the moment of inertia. 1.9 Expression for Torque in Terms of Moment of Inertia: Fig: Expression for torque A rigid object rotating with a constant angular acceleration α about an axis perpendicular to the plane of paper. Let us consider the object to be consisting of N number of particles of masses 𝒎𝟏,𝒎𝟐, … … . . , 𝒎𝑵 at respective perpendicular distances 𝒓𝟏, 𝒓𝟐,… … . . , 𝒓𝑵 from the axis of rotation. Linear accelerations 𝒂𝟏 = 𝒓𝟏𝜶, 𝒂𝟐 = 𝒓𝟐𝜶,………., 𝒂𝑵 = 𝒓𝑵𝜶 etc. Force experienced by the first particle is 𝒇𝟏 = 𝒎𝟏𝒂𝟏 = 𝒎𝟏𝒓𝟏𝜶 As these forces are tangential, their respective perpendicular distances from the axis are 𝒓𝟏, 𝒓𝟐,… . . 𝒓𝑵. Thus, the torque experienced by the first particle is of magnitude 𝝉𝟏 = 𝒇𝟏𝒓𝟏 = 𝒎𝟏𝒓𝟏 𝟐 𝜶 Similarly, 𝝉𝟐 = 𝒎𝟐𝒓𝟐 𝟐 𝜶, 𝝉𝟑 = 𝒎𝟑𝒓𝟑 𝟐 𝜶,………, 𝝉𝑵 = 𝒎𝑵𝒓𝑵 𝟐 𝜶 Magnitude of the resultant torque is,
11 where, I = 𝒎𝟏𝒓𝟏 𝟐 + 𝒎𝟐𝒓𝟐 𝟐 … … . . +𝒎𝑵𝒓𝑵 𝟐 is the moment of inertia of the object about the given axis of rotation. The relation 𝝉 = 𝑰𝜶 is analogous to f = ma for the translational motion if the moment of inertia I replaces mass, which is its physical significance. 1.10 Conservation of Angular Momentum: Angular momentum of a system is, 𝑳 ⃗ ⃗ = 𝒓 ⃗ × 𝒑 ⃗ ⃗ Where, 𝒓 ⃗ is the position vector from the axis of rotation and 𝒑 ⃗ ⃗ is the linear momentum. Hence, angular momentum 𝑳 ⃗ ⃗ is conserved in the absence of external unbalanced torque 𝝉 ⃗ . This is the principle of conservation of angular momentum. Examples of conservation of angular momentum: During some shows of ballet dance, acrobat in a circus, sports like ice skating, diving in a swimming pool, etc., the principle of conservation of angular momentum is realized. In all these applications the product L = Iw = I (2𝝅𝒏) is constant. (i) Ballet dancers: During ice ballet, the dancers have to undertake rounds of smaller and larger radii. The dancers come together while taking the rounds of smaller radius. The moment of inertia of their system becomes minimum and the frequency increases, to make it thrilling. While outer rounds, the dancers outstretch their legs and arms. This increases their moment of inertia that reduces the angular speed and hence the linear speed. This is essential to prevent slipping. (ii) Diving in a swimming pool (during competition): While on the diving board, the divers stretch their body so as to increase the moment of inertia. Immediately after leaving the board, they fold their body. This reduces the moment inertia considerably. As a result, the frequency increases and they can complete more rounds in air to make the show attractive. Again, while entering into water they stretch their body into a streamline shape. This allows them a smooth entry into the water. 1.11 Rolling Motion: The objects like a cylinder, sphere, wheels, etc. are quite often seen to perform rolling motion. (i) circular motion of the body as a whole, about its own symmetric axis and (ii) linear motion of the body assuming it to be concentrated at its centre of mass. Consider an object of moment of inertia I, rolling uniformly. v = Linear speed of the centre of mass, R = Radius of the body, w = Angular speed of rotation ∴ 𝒘 = 𝑽 𝑹 for any particle M = Mass of the body, K = Radius of gyration of the body ∴ 𝑰 = 𝑴𝑲𝟐 Total kinetic energy of rolling = Translational K.E. + Rotational K.E.
12 Static friction is essential for a purely rolling motion. It prevents the sliding motion. 1.11.1 Linear Acceleration and Speed While Pure Rolling Down an Inclined Plane: Inclination of the plane with the horizontal is θ. Fig: Rolling along an incline As the objects starts rolling down, its gravitational P.E. is converted into K.E. of rolling. Linear distance travelled along the plane is s = 𝒉 𝒔𝒊𝒏 𝜽 During this distance, the linear velocity has increased from zero to v. For pure sliding, without friction, the acceleration is g sin θ and final velocity is √𝟐𝒈𝒉 . Thus, during pure rolling, the factor (1 + 𝑲𝟐 𝑹𝟐) is effective for both the expressions.

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PHYSICS CLASS XII Chapter 1 - Rotationall dynamics

  • 1.
  • 2.
    2 1.1 INTRODUCTION Every daywe come across several revolving or rotating (rigid) objects. During revolution, the object undergoes circular motion about some point outside the object or about some other object, while during rotation the motion is about an axis of rotation passing through the object. 1.2 Characteristics of Circular Motion: 1) It is an accelerated motion 2) It is a periodic motion 1.2.1 Kinematics of Circular Motion: The tangential velocity is, 𝒗 ⃗ ⃗ = 𝒘 ⃗⃗⃗ x 𝒓 ⃗ , where 𝒘 ⃗⃗⃗ is the angular velocity. The magnitude of 𝒗 ⃗ ⃗ is v = ω r. Direction of 𝒘 ⃗⃗⃗ is always along the axis of rotation and is given by the right-hand thumb rule. Fig. Directions of angular velocity If T is period of circular motion and n is the frequency, w = 2𝝅n = 𝟐𝝅 𝑻 Uniform circular motion: During circular motion if the speed of the particle remains constant, it is called Uniform Circular Motion (UCM). The acceleration responsible for this is the centripetal or radial acceleration 𝒂𝒓 ⃗⃗⃗⃗ = - 𝒘𝟐 𝒓 ⃗ For UCM, its magnitude is constant and it is a = 𝒘𝟐 r = 𝒗𝟐 𝒓 = vw. It is always directed towards the centre of the circular motion, hence called centripetal. Fig. Directions of linear velocity and acceleration. Non-uniform circular motion: For non-uniform circular motion, the magnitude of 𝒂𝒓 ⃗⃗⃗⃗ is not constant. This is due to the angular acceleration 𝜶 ⃗⃗ = 𝒅𝒘 ⃗⃗ ⃗ 𝒅𝒕 For increasing speed, it is along the direction of 𝒘 ⃗⃗⃗ while during decreasing speed, it is opposite to that of 𝒘 ⃗⃗⃗ . Fig. Direction of angular acceleration. If the angular acceleration 𝜶 ⃗⃗ is constant and along the axis of rotation, all 𝜽 ⃗ ⃗ , 𝒘 ⃗⃗⃗ and 𝜶 ⃗⃗ will be directed along the axis. 1.2.2 Dynamics of Circular Motion (Centripetal Force and Centrifugal Force):
  • 3.
    3 i) Centripetal force(CPF): The acceleration responsible for circular motion is the centripetal or radial acceleration 𝒂𝒓 ⃗⃗⃗⃗ = - 𝒘𝟐 𝒓 ⃗ . The force providing this acceleration is the centripetal or radial force, CPF = −𝒎𝒘𝟐 𝒓 ⃗ ii) Centrifugal force (c.f.f.): This force, (+𝒎𝒘𝟐 𝒓 ⃗ ) the centrifugal force. It is a pseudo force arising due to the centripetal acceleration of the frame of reference. It must be understood that centrifugal force is a non-real force, but NOT an imaginary force. Resultant force = −𝒎𝒘𝟐 𝒓 ⃗ Or 𝒎𝒘𝟐 𝒓 ⃗ + ∑(𝒓𝒆𝒂𝒍 𝒇𝒐𝒓𝒄𝒆𝒔) = 0 1.3 Applications of Uniform Circular Motion: 1.3.1 Vehicle Along a Horizontal Circular Track: Forces acting on the car (considered to be a particle) are (i) weight mg, vertically downwards, (ii) normal reaction N, vertically upwards that balances the weight mg and (iii) force of static friction 𝒇𝒔 between road and the tyres. This is static friction because it prevents the vehicle from outward slipping or skidding. This is the resultant force which is centripetal. Fig. Vehicle on a horizontal road ∴ 𝒎𝒈 = 𝑵 𝒂𝒏𝒅 𝒇𝒔 = 𝒎𝒓𝒘𝟐 = 𝒎𝒗𝟐 𝒓 ∴ 𝒇𝒔 𝑵 = 𝒓𝒘𝟐 𝒈 = 𝒗𝟐 𝒓𝒈 For a given track, radius r is constant. For given vehicle, mg = N is constant. (𝒇𝒔)𝒎𝒂𝒙 = 𝝁𝒔. 𝑵, where, 𝝁𝒔 is the coefficient of static friction At the maximum possible speed 𝒗𝒔, (𝒇𝒔)𝒎𝒂𝒙 𝑵 = 𝝁𝒔 = 𝒗𝒎𝒂𝒙 𝟐 𝒓𝒈 ∴ 𝒗𝒎𝒂𝒙 = √𝝁𝒔𝒓𝒈 1.3.2 Well (or Wall) of Death: (मौत का क ु आँ): The forces acting on the vehicle are (i) Normal reaction N acting horizontally and towards the centre, (ii) Weight mg acting vertically downwards, and (iii) Force of static friction 𝒇𝒔 acting vertically upwards between vertical wall and the tyres. Fig. Well of death N = mr𝒘𝟐 = 𝒎𝒗𝟐 𝒓 and mg = 𝒇𝒔 Force of static friction 𝒇𝒔 is always less than or equal to 𝝁𝒔N. ∴ 𝒇𝒔 ≤ 𝝁𝒔 ( 𝒎𝒗𝟐 𝒓 )
  • 4.
    4 ∴ 𝒈 ≤ 𝝁𝒔𝒗𝟐 𝒓 ∴𝒗𝟐 ≥ 𝒓𝒈 𝝁𝒔 ∴ 𝑽𝒎𝒊𝒏 = √ 𝒓𝒈 𝝁𝒔 1.3.3 Vehicle on a Banked Road: “Defined as the phenomenon in which the edge is raised for the curved roads above the inner edge to provide the necessary centripetal force to the vehicle to that they take a safe turn” There are two forces acting on the vehicle, (i) weight mg, vertically downwards and (ii) normal reaction N, perpendicular to the surface of the road. Its vertical component Ncosθ balances weight mg. Horizontal component Nsinθ being the resultant force, must be the necessary centripetal force. Fig: Vehicle on a banked road. N cos 𝜽 = mg and N sin 𝜽 = mr𝒘𝟐 = 𝒎𝒗𝟐 𝒓 ∴ 𝒕𝒂𝒏 𝜽 = 𝒗𝟐 𝒓𝒈 (a) Most safe speed: For a particular road, r and θ are fixed. 𝒗𝒔 = √𝒓𝒈 𝒕𝒂𝒏 𝜽 (b) Banking angle: While designing a road, this expression helps us in knowing the angle of banking as 𝜽 = 𝒕𝒂𝒏−𝟏 ( 𝒗𝟐 𝒓𝒈 ) (c) Speed limits: If the vehicle is running exactly at the speed 𝒗𝒔 = √𝒓𝒈 𝒕𝒂𝒏 𝜽, the forces acting on the vehicle are (i) weight mg acting vertically downwards and (ii) normal reaction N acting perpendicular to the road. Fig: Banked road: lower speed limit For speeds 𝒗𝟏 < √𝒓𝒈 𝒕𝒂𝒏 𝜽, 𝒎𝒗𝟏 𝟐 𝒓 < 𝑵 𝒔𝒊𝒏 𝜽.
  • 5.
    5 Fig: Banked road:upper speed limit. These two forces take care of the necessary centripetal force. ∴ 𝒎𝒈 = 𝒇𝒔 𝒔𝒊𝒏 𝜽 + 𝑵 𝒄𝒐𝒔 𝜽 and For minimum possible speed, 𝒇𝒔 is maximum and equal to 𝝁𝒔N. For 𝝁𝒔 ≥ tan 𝜽, 𝒗𝒎𝒊𝒏 = 0. This is true for most of the rough roads, banked at smaller angles. (d) For speeds 𝒗𝟐 = √𝒓𝒈 𝒕𝒂𝒏 𝜽, 𝒎𝒗𝟐 𝟐 𝒓 > 𝑵 𝒔𝒊𝒏 𝜽. For maximum possible speed, 𝒇𝒔 is maximum and equal to 𝝁𝒔N. If 𝝁𝒔 = cot θ, 𝒗𝒎𝒂𝒙 = ∞. But (𝝁𝒔)𝒎𝒂𝒙 = 1. Thus, for θ ≥ 45°, 𝒗𝒎𝒂𝒙 = ∞. (e) For 𝝁𝒔 = 0, both the equations give us v = √𝒓𝒈 𝒕𝒂𝒏 𝜽 which is the safest speed 1.3.4 Conical Pendulum: A tiny mass connected to a long, flexible, massless, inextensible string, and suspended to a rigid support is called a pendulum. The string moves along the surface of a right circular cone of vertical axis and the point object performs a (practically) uniform horizontal circular motion is called a conical pendulum. Fig: In an inertial frame The forces acting on the bob are (i) its weight mg directed vertically downwards and (ii) the force 𝑻𝟎 due to the tension in the string, directed along the string, towards the support A.
  • 6.
    6 Fig: In anon- inertial frame Radius r of the circular motion is r = L sin 𝜽. 1.4 Vertical Circular Motion: Two types of vertical circular motions are: (a) A controlled vertical circular motion such as a giant wheel or similar games. In this case the speed is either kept constant or NOT totally controlled by gravity. (b) Vertical circular motion controlled only by gravity. In this case, we initially supply the necessary energy (mostly) at the lowest point. 1.4.1 Point Mass Undergoing Vertical Circular Motion Under Gravity: Case I: Mass tied to a string: Both, weight (mg) and force due to tension are downward i.e., towards the center. There are only resultant centripetal force acts towards centre. Fig: Vertical circular motion Uppermost position (A): Thus, if 𝑽𝑨 is the speed at the uppermost point, Radius r of the circular motion is the length of the string. For minimum possible speed, Lowermost position (B): Force due to the tension, 𝑻𝑩 is vertically upwards, and opposite to mg. If 𝑽𝑩 is the speed at the lowermost point, The vertical displacement is 2r and the motion is governed only by gravity.
  • 7.
    7 Positions when thestring is horizontal (C and D): Force due to the tension is the only force towards the centre as weight mg is perpendicular to the tension. Arbitrary positions: Force due to the tension and weight are neither along the same line, nor perpendicular. It decreases the speed while going up and increases it while coming down. Case II: Mass tied to a rod: Consider a bob tied to a rod and whirled along a vertical circle. Zero speed is possible at the uppermost point. 1.4.2 Sphere of Death (मृत्यु गोल): Two-wheeler rider undergo rounds inside a hollow sphere. Starting with small horizontal circles, they eventually perform revolutions along vertical circles. The dynamics of this vertical circular motion is the same as that of the point mass tied to the string, except that the force due to tension T is replaced by the normal reaction force N. The linear speed is more for larger circles but angular speed is more for smaller circles. This is as per the theory of conical pendulum. 1.4.3 Vehicle at the Top of a Convex Over-Bridge: Fig: Vehicle on a convex over-bridge Forces acting on the vehicle are (a) Weight mg and (b) Normal reaction force N, both along the vertical line. Thus, if v is the speed at the uppermost point, As the speed is increased, N goes on decreasing. Thus, for just maintaining contact, N = 0. This imposes an upper limit on the speed as 𝒗𝒎𝒂𝒙 = √𝒓𝒈 1.5 Moment of Inertia as an Analogous Quantity for Mass: Fig: A body of N particles Let us consider the object to be consisting of N particles of masses 𝒎𝟏, 𝒎𝟐 .………𝒎𝒏 at respective perpendicular distances 𝒓𝟏,𝒓𝟐,… … … 𝒓𝒏 from the axis of rotation. Translational K.E. of the first particle is
  • 8.
    8 Thus, rotational K.E. IfI = ∑ 𝒎𝒊𝒓𝒊 𝟐 replaces mass m and angular speed ω replaces linear speed v, rotational K.E. = 𝟏 𝟐 𝑰𝒘𝟐 is analogous to translational K.E. = 𝟏 𝟐 𝒎𝒗𝟐 Thus, I is defined to be the rotational inertia or moment of inertia (M.I.) The moment of inertia of an object depends upon (i) individual masses and (ii) the distribution of these masses about the given axis of rotation. The moment of inertia is to be obtained by integration as I = ∫ 𝒓𝟐 𝒅𝒎 . Fig: Moment of Inertia of a ring 1.5.1 Moment of Inertia of a Uniform Ring: An object is called a uniform ring if its mass is situated uniformly on the circumference of a circle. If it is rotating about its own axis, its entire mass M is practically at a distance equal to its radius R form the axis. The moment of inertia of a uniform ring of mass M and radius R is I = 𝑴𝑹𝟐 . 1.5.2 Moment of Inertia of a Uniform Disc: The ratio 𝝈 = 𝒎 𝑨 = 𝒎𝒂𝒔𝒔 𝒂𝒓𝒆𝒂 is called the surface density. Consider a uniform disc of mass M and radius R rotating about its own axis, which is the line perpendicular to its plane and passing through its centre 𝝈 = 𝑴 𝝅𝑹𝟐. Fig: Moment of Inertia of a disk Width of this ring is dr, which is so small that the entire ring can be considered to be of average radius r. Area of this ring is A = 2πr.dr ∴ 𝝈 = 𝒅𝒎 𝟐𝝅𝒓.𝒅𝒓 ∴ dm = 2πσr.dr. The moment of inertia of this ring is 𝑰𝒓 = 𝒅𝒎 (𝒓𝟐 ) Moment of inertia (I) of the disc can now be obtained by integrating 𝑰𝒓 from r = 0 to r = R.
  • 9.
    9 1.6 Radius ofGyration: It depends upon mass of that object and how that mass is distributed around the axis of rotation. The moment of inertia of any object as I = 𝑴𝑲𝟐 , where M is mass of that object. K is defined as the radius of gyration of the object about the given axis of rotation, I = 𝑴𝑲𝟐 . Larger the value of K, farther is the mass from the axis. 1.7 Theorem of Parallel Axes and Theorem of Perpendicular Axes: Expressions of moment of inertias of regular geometrical shapes are about their axes of symmetry. 1.7.1 Theorem of Parallel Axes: Fig: Theorem of parallel axes Consider, MOP is any axis passing through point O. Axis ACB is passing through the centre of mass C of the object, parallel to the axis MOP, and at a distance h from it (∴ h = CO). Consider a mass element dm located at point D. Perpendicular on OC from point D is DN. Moment of inertia of the object about the axis ACB is 𝑰𝑪 = ∫(𝑫𝑪)𝟐 𝒅𝒎, and about the axis MOP it is 𝑰𝟎 = ∫(𝑫𝟎)𝟐 𝒅𝒎 . From the definition of the centre of mass, ∫ 𝑵𝑪. 𝒅𝒎 = 𝟎 . It states that, “The moment of inertia (𝑰𝟎) of an object about any axis is the sum of its moment of inertia (𝑰𝑪) about an axis parallel to the given axis, and passing through the centre of mass and the product of the mass of the object and the square of the distance between the two axes (M𝒉𝟐 ).” 1.7.2 Theorem of Perpendicular Axes: Consider, 𝑰𝒙 and 𝑰𝒚 be the moment of Inertia of axis passing through point ‘M’ and ‘N’. Fig: Theorem of perpendicular axes Consider a mass element dm located at any point P. PM = y and PN = x are the perpendiculars drown from P respectively on the x and y axes.
  • 10.
    10 z = √𝒚𝟐+ 𝒙𝟐. If 𝑰𝒙, 𝑰𝒚 and 𝑰𝒛 are the respective moment of inertias It states that, “The moment of inertia (𝑰𝒛) of a laminar object about an axis (z) perpendicular to its plane is the sum of its moment of inertias about two mutually perpendicular axes (x and y) in its plane, all the three axes being concurrent”. 1.8 Angular Momentum or Moment of Linear Momentum: 𝑳 ⃗ ⃗ = 𝒓 ⃗ × 𝒑 ⃗ ⃗ , were 𝒓 ⃗ is the position vector from the axis of rotation. In magnitude, L = P × r sin θ, where θ is the smaller angle between the directions of 𝑷 ⃗⃗ and 𝒓 ⃗ . 1.8.1 Expression for Angular Momentum in Terms of Moment of Inertia: A rigid object rotating with a constant angular speed ω about an axis perpendicular to the plane of paper. Let us consider the object to be consisting of N number of particles of masses 𝒎𝟏, 𝒎𝟐,… … … . 𝒎𝑵 at respective perpendicular distances 𝒓𝟏, 𝒓𝟐,… … … . 𝒓𝑵 from the axis of rotation. As the object rotates, all these particles perform UCM with same angular speed ω, but with different linear speeds 𝒗𝟏 = 𝒓𝟏𝒘, 𝒗𝟐 = 𝒓𝟐𝒘,… … … . 𝒗𝑵 = 𝒓𝑵𝒘. Directions of individual velocities 𝒗𝟏 ⃗⃗⃗⃗ , 𝒗𝟐 ⃗⃗⃗⃗ , etc., are along the tangents to their respective tracks. Linear momentum of the first particle is of magnitude 𝒑𝟏 = 𝒎𝟏𝒗𝟏 = 𝒎𝟏𝒓𝟏𝒘. Its direction is along that of 𝒗𝟏 ⃗⃗⃗⃗ . Its angular momentum is thus of magnitude 𝑳𝟏 = 𝒑𝟏𝒓𝟏 = 𝒎𝟏𝒓𝟏 𝟐 𝒘 Similarly, 𝑳𝟐 = 𝒎𝟐𝒓𝟐 𝟐 𝒘, 𝑳𝟑 = 𝒎𝟑𝒓𝟑 𝟐 𝒘, ………., 𝑳𝑵 = 𝒎𝑵𝒓𝑵 𝟐 𝒘 Thus, magnitude of angular momentum of the body is, Where, I = 𝒎𝟏𝒓𝟏 𝟐 + 𝒎𝟐𝒓𝟐 𝟐 + …. + 𝒎𝑵𝒓𝑵 𝟐 is the moment of inertia. 1.9 Expression for Torque in Terms of Moment of Inertia: Fig: Expression for torque A rigid object rotating with a constant angular acceleration α about an axis perpendicular to the plane of paper. Let us consider the object to be consisting of N number of particles of masses 𝒎𝟏,𝒎𝟐, … … . . , 𝒎𝑵 at respective perpendicular distances 𝒓𝟏, 𝒓𝟐,… … . . , 𝒓𝑵 from the axis of rotation. Linear accelerations 𝒂𝟏 = 𝒓𝟏𝜶, 𝒂𝟐 = 𝒓𝟐𝜶,………., 𝒂𝑵 = 𝒓𝑵𝜶 etc. Force experienced by the first particle is 𝒇𝟏 = 𝒎𝟏𝒂𝟏 = 𝒎𝟏𝒓𝟏𝜶 As these forces are tangential, their respective perpendicular distances from the axis are 𝒓𝟏, 𝒓𝟐,… . . 𝒓𝑵. Thus, the torque experienced by the first particle is of magnitude 𝝉𝟏 = 𝒇𝟏𝒓𝟏 = 𝒎𝟏𝒓𝟏 𝟐 𝜶 Similarly, 𝝉𝟐 = 𝒎𝟐𝒓𝟐 𝟐 𝜶, 𝝉𝟑 = 𝒎𝟑𝒓𝟑 𝟐 𝜶,………, 𝝉𝑵 = 𝒎𝑵𝒓𝑵 𝟐 𝜶 Magnitude of the resultant torque is,
  • 11.
    11 where, I =𝒎𝟏𝒓𝟏 𝟐 + 𝒎𝟐𝒓𝟐 𝟐 … … . . +𝒎𝑵𝒓𝑵 𝟐 is the moment of inertia of the object about the given axis of rotation. The relation 𝝉 = 𝑰𝜶 is analogous to f = ma for the translational motion if the moment of inertia I replaces mass, which is its physical significance. 1.10 Conservation of Angular Momentum: Angular momentum of a system is, 𝑳 ⃗ ⃗ = 𝒓 ⃗ × 𝒑 ⃗ ⃗ Where, 𝒓 ⃗ is the position vector from the axis of rotation and 𝒑 ⃗ ⃗ is the linear momentum. Hence, angular momentum 𝑳 ⃗ ⃗ is conserved in the absence of external unbalanced torque 𝝉 ⃗ . This is the principle of conservation of angular momentum. Examples of conservation of angular momentum: During some shows of ballet dance, acrobat in a circus, sports like ice skating, diving in a swimming pool, etc., the principle of conservation of angular momentum is realized. In all these applications the product L = Iw = I (2𝝅𝒏) is constant. (i) Ballet dancers: During ice ballet, the dancers have to undertake rounds of smaller and larger radii. The dancers come together while taking the rounds of smaller radius. The moment of inertia of their system becomes minimum and the frequency increases, to make it thrilling. While outer rounds, the dancers outstretch their legs and arms. This increases their moment of inertia that reduces the angular speed and hence the linear speed. This is essential to prevent slipping. (ii) Diving in a swimming pool (during competition): While on the diving board, the divers stretch their body so as to increase the moment of inertia. Immediately after leaving the board, they fold their body. This reduces the moment inertia considerably. As a result, the frequency increases and they can complete more rounds in air to make the show attractive. Again, while entering into water they stretch their body into a streamline shape. This allows them a smooth entry into the water. 1.11 Rolling Motion: The objects like a cylinder, sphere, wheels, etc. are quite often seen to perform rolling motion. (i) circular motion of the body as a whole, about its own symmetric axis and (ii) linear motion of the body assuming it to be concentrated at its centre of mass. Consider an object of moment of inertia I, rolling uniformly. v = Linear speed of the centre of mass, R = Radius of the body, w = Angular speed of rotation ∴ 𝒘 = 𝑽 𝑹 for any particle M = Mass of the body, K = Radius of gyration of the body ∴ 𝑰 = 𝑴𝑲𝟐 Total kinetic energy of rolling = Translational K.E. + Rotational K.E.
  • 12.
    12 Static friction isessential for a purely rolling motion. It prevents the sliding motion. 1.11.1 Linear Acceleration and Speed While Pure Rolling Down an Inclined Plane: Inclination of the plane with the horizontal is θ. Fig: Rolling along an incline As the objects starts rolling down, its gravitational P.E. is converted into K.E. of rolling. Linear distance travelled along the plane is s = 𝒉 𝒔𝒊𝒏 𝜽 During this distance, the linear velocity has increased from zero to v. For pure sliding, without friction, the acceleration is g sin θ and final velocity is √𝟐𝒈𝒉 . Thus, during pure rolling, the factor (1 + 𝑲𝟐 𝑹𝟐) is effective for both the expressions.