Rotational Motion: Centre of Mass
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Questions and Answers

Two particles of mass 2 kg and 4 kg move on a linear path in opposite directions with velocities 2 m/s and 3 m/s, respectively. What is the velocity of the center of mass of the system?

3 m/s in the direction of the first object

Due to their gravitational forces, the acceleration produced in object m2, located at distances r1 and r2 from the center of mass of the system, is?

  • r2^2 (correct)
  • (r1 – r2)^2
  • (r1 * r2)^2
  • (r1 * r2)^2

    • When a stone of mass 90 kg is pulled by a person of mass 60 kg on a frictionless surface, they will meet each other at a distance of ______ meters from the person.

    5m

    What is the unit of instantaneous angular acceleration?

    <p>radians per second squared</p> Signup and view all the answers

    Define the moment of inertia for a particle.

    <p>Moment of inertia for a particle is defined as I = mr^2, where r is the perpendicular distance of the particle from the axis.</p> Signup and view all the answers

    The theorem of parallel axis states that I = Ic + Md^2, where I is the moment of inertia of the body about any axis, Ic is ______, M is total mass of the substance, and d is the perpendicular distance between the two axes.

    <p>Moment of inertia of the body through its center about any axis</p> Signup and view all the answers

    Torque is a vector quantity.

    <p>True</p> Signup and view all the answers

    Match the following units with their correct representation:

    <p>Nm = Newton meter kg m2 = Moment of inertia SI unit J s = Angular momentum unit cm = Centimeter</p> Signup and view all the answers

    Three particles each of mass 3 kg are placed at three corners of an equilateral triangle. The centre of mass with respect to particle 1 is?

    <p>(1.33, 0.5) meters</p> Signup and view all the answers

    Two particles of mass 50 g and 100 g have positions (3i + 4j + 5k) cm and (-6i - 2k + 4j) cm with respect to the origin. The distance of the centre of mass from the origin is?

    <p>15 cm</p> Signup and view all the answers

    Find the centre of mass with respect to the origin of an E shape having 2 cm thickness and uniform density distribution.

    <p>(2.4, 5) cm</p> Signup and view all the answers

    Find the centre of mass with respect to a particle of 1g mass of a four-particle parallelogram-shaped system.

    <p>(0.95a, 3a)</p> Signup and view all the answers

    At which point from the surface should a force be applied to a 'T'-shaped object so that it has only translation motion?

    <p>2l</p> Signup and view all the answers

    Four bricks each of length L and mass m are arranged as shown. The distance of the centre of mass of the system from the wall is?

    <p>8L</p> Signup and view all the answers

    A circular plate of uniform thickness has a diameter of 60 cm. The centre of mass of the remaining portion with respect to the origin is?

    <p>(0, -8) cm</p> Signup and view all the answers

    The particles of 10g, 20g, 30g, and 40g are placed at 2, 6, 8, and 11-hour symbols respectively of a weightless clock dial of radius 8cm. Find the coordinates of the centre of mass of this system.

    <p>(1.49, -0.184) cm</p> Signup and view all the answers

    The distance of the centre of mass from x=0 of a rod of length L changes linearly with x according to the equation l = bx. The distance of the centre of mass from x=0 is?

    <p>2/3L</p> Signup and view all the answers

    The centre of mass of a half portion of a thin ring of mass 2M and radius R having uniform mass density with respect to its centre is?

    <p>2SR</p> Signup and view all the answers

    Which equation represents the Law of conservation of angular momentum?

    <p>L = constant</p> Signup and view all the answers

    If a planet comes near to the sun, what happens to its moment of inertia and angular speed?

    <p>Moment of inertia decreases, angular speed increases</p> Signup and view all the answers

    What is the formula for rotational kinetic energy?

    <p>KR = 1/2 Iw^2</p> Signup and view all the answers

    The work due to torque is given by the formula W = ______.

    <p>∫ τ dθ</p> Signup and view all the answers

    If a ball of mass m and radius r is hit at height h from its center with a velocity gained v0, then the angular speed obtained by it is?

    <p>2/5</p> Signup and view all the answers

    In a system of two rods of mass m and length l each placed perpendicular to each other, the moment of inertia about x x' axis is?

    <p>6ml^2</p> Signup and view all the answers

    Match the following formulas with their corresponding concepts:

    <p>8MR^2 = Moment of inertia of a system of discs cut from a square plate 3MR^2/8 = Moment of inertia of a remaining portion of a square plate after cutting four discs of radius R 3/8MR^2 = Moment of inertia of a square plate of side 4R with four discs cut out</p> Signup and view all the answers

    The increase in angular speed of a turn table after firing a bullet of mass m with speed v in the opposite direction of the table's motion is?

    <p>mvr/I0</p> Signup and view all the answers

    The height of point C from the center of the rod where a colliding particle comes to a stop in order for end A of the rod to remain stationary is ______.

    <p>L/2</p> Signup and view all the answers

    The moment of inertia of a wheel about an axis perpendicular to its plane is 2.5 kgm^2.

    <p>False</p> Signup and view all the answers

    Which of the following objects will reach the bottom of an inclined plane first?

    <p>Solid sphere</p> Signup and view all the answers

    A hollow sphere of mass M and radius R rolls on a horizontal plane without slipping. If the surface rolls up vertically, what will be the maximum height the body would attain?

    <p>$10\frac{v^2}{7g}$</p> Signup and view all the answers

    A small object of uniform density rolls up a curved surface with an initial velocity v. It reaches a maximum height of 4g with respect to the initial position. What type of object is it?

    <p>Hollow sphere</p> Signup and view all the answers

    On a frictionless inclined plane, a solid cylinder is rolling down. What is the relationship between the frictional force and the angle of inclination?

    <p>Frictional force opposes linear motion and helps rotational motion</p> Signup and view all the answers

    A small sphere of radius r is kept in a hemispherical bowl of radius R. If the sphere is released from a point A, what will be its angular speed at the bottom of the bowl?

    <p>$7\frac{r}{10R}\sqrt{g}$</p> Signup and view all the answers

    The kinetic energy of the body at the bottom of an inclined plane is more in which option?

    <p>Information is incomplete</p> Signup and view all the answers

    For a solid sphere and a solid cylinder with equal mass rolling down on inclined planes, what should be the ratio of heights for the objects to have equal velocities at the bottom?

    <p>14:5</p> Signup and view all the answers

    On an inclined plane with heights 2h, what is the ratio of linear kinetic energy to rotational kinetic energy of a solid sphere at a certain point?

    <p>6</p> Signup and view all the answers

    In what condition will a light coin placed at a distance r from the center of a rotating turn table keep rotating with the turn table?

    <p>r ≤ Pg</p> Signup and view all the answers

    A particle moving on a straight line with uniform velocity always has an angular momentum of zero.

    <p>True</p> Signup and view all the answers

    A hollow cylinder of metal reaches the bottom of an inclined plane before a solid cylinder of wood because of unequal kinetic energy.

    <p>True</p> Signup and view all the answers

    Inertia and moment of inertia represent the same quantities.

    <p>False</p> Signup and view all the answers

    At a high position on a ladder, torque is larger compared to when just starting to climb.

    <p>True</p> Signup and view all the answers

    When there is no torque acting on a body's center of mass, its speed remains constant.

    <p>True</p> Signup and view all the answers

    A wheel moving down a frictionless inclined plane can only slip and cannot roll.

    <p>False</p> Signup and view all the answers

    A hollow cylinder is stronger than a solid cylinder when used as a shaft in a motor due to torque differences.

    <p>True</p> Signup and view all the answers

    The linear speed for all particles of a rolling body is constant.

    <p>True</p> Signup and view all the answers

    If the Earth halves in size (with constant mass), the length of the day decreases.

    <p>True</p> Signup and view all the answers

    The value of radius of gyration is constant for a given body.

    <p>False</p> Signup and view all the answers

    How does the angular velocity of a rotating circular platform vary with time as a child moves along a chord?

    <p>w0, w0, w0, 0</p> Signup and view all the answers

    Match the columns for different scenarios involving a projectile's motion.

    <p>Torque acting on the body = Angular momentum of the body 25 SI = 50 SI i ® Q ii ® R iii ® R iv ® R = i ® R ii ® Q iii ® P iv ® R P = 0 = Q = constant</p> Signup and view all the answers

    Study Notes

    Centre of Mass

    • A point at which all the mass of the system can be considered as concentrated.
    • Defined for the study of extended objects as a particle.
    • Centre of mass can be inside or outside the body.
    • Shows the average position of the mass of the component of the object.

    Centre of Mass for Two Particles

    • M r cm = m1 r1 + m2 r2 where M = m1 + m2
    • In component form, M xcm = m1x1 + m2x2 and M ycm = m1y1 + m2y2
    • If centre of mass is at origin, m1 r1 + m2 r2 = 0
    • Sign of r1 and r2 are opposite, showing that the mass m1 and m2 are on both sides of the centre of mass.

    Centre of Mass for n-Particles System

    • M r cm = ¦ mi ri where i = 1 to n
    • In component form, Mxcm = ¦ mi xi, Mycm = ¦ mi yi and Mzcm = ¦ mi zi
    • For a rigid body, M r cm = ³ r dm, Mxcm = ³ x dm, Mycm = ³ y dm and Mzcm = ³ z dm

    Motion of Centre of Mass

    • M v cm = ¦ mi vi where i = 1 to n
    • F net = M a cm = Resultant external force
    • Linear momentum of the system: P = M v cm
    • Conservation of linear momentum: If F ext = 0 then dP/dt = 0, and P = constant

    Problems and Solutions

    • 15 multiple-choice questions with solutions related to centre of mass, motion of centre of mass, and linear momentum.
    • Questions involve various scenarios, such as particles, rods, spheres, and blocks, with different masses and motions.### Rotational Motion
    • Angular displacement (θ) is a vector quantity, measured in radians or revolutions.
    • Its direction can be found using the right-hand screw rule.
    • Angular velocity (ω) is the rate of change of angular displacement.
    • Average angular velocity (ω_avg) = Δθ / Δt.
    • Instantaneous angular velocity (ω) = dθ / dt.
    • Unit of angular velocity is rad/s.
    • Angular acceleration (α) is the rate of change of angular velocity.
    • Average angular acceleration (α_avg) = Δω / Δt.
    • Instantaneous angular acceleration (α) = dω / dt.
    • Unit of angular acceleration is rad/s².
    • Relation between angular velocity and linear velocity: v = ω × r.
    • Relation between angular acceleration and linear acceleration: a = α × r + ω × v.

    Angular Displacement and Angular Velocity

    • Angular displacement (θ) = r × Δφ.
    • Angular velocity (ω) = dθ / dt.
    • For a fixed axis of rotation, angular displacement is different.
    • Angular velocity is an axial vector, so its direction is along the rotational axis.

    Angular Acceleration

    • Angular acceleration (α) = dω / dt.
    • Unit of angular acceleration is rad/s².
    • Direction of angular acceleration is in the direction of change in angular velocity.

    Relations with Linear Motion

    • Relation between angular velocity and linear velocity: v = ω × r.
    • Relation between angular acceleration and linear acceleration: a = α × r + ω × v.
    • Tangential component of acceleration (a_T) = α × r.
    • Radial component of acceleration (a_r) = ω × v.

    Moment of Inertia

    • Moment of inertia (I) is a characteristic of an object that opposes the change of motion.
    • For a particle, I = mr².
    • For a system of particles, I = Σm_i r_i².
    • For a uniform distribution of mass, I = ∫r² dm.
    • SI unit of moment of inertia is kg m².
    • Dimensional formula is M¹L²T⁰.
    • Moment of inertia depends on the selection of axis and the distribution of mass about it.
    • Moment of inertia is a tensor physical quantity.

    Radius of Gyration

    • Radius of gyration (k) is the perpendicular distance from the axis of a particle at which the total mass of the object is concentric.
    • k = √(r1² + r2² + ... + rn²) / n.
    • Moment of inertia (I) = Mk².

    Theorem of Parallel Axis

    • I = I_c + Md².
    • I_c is the moment of inertia of the body through its center about any axis.
    • M is the total mass of the substance.
    • d is the perpendicular distance between the two axes.

    Theorem of Perpendicular Axis

    • Iz = Ix + Iy.
    • This theorem is applicable for a planar body.

    Torque

    • Torque (τ) is the moment of force with respect to a given reference point.
    • τ = r × F.
    • Unit of torque is Nm or J.
    • Dimensional formula is M¹L²T⁻².
    • Torque is a vector quantity, and its direction can be found using the right-hand screw rule.

    Couple

    • A couple is a system of two forces of equal magnitude and opposite directions that are not collinear.
    • Moment of couple = magnitude of any one of the two forces × perpendicular distance between the two forces.

    Angular Momentum

    • Angular momentum (L) = r × P.
    • L = rpsinθ.
    • Unit of angular momentum is J s or erg s.
    • Angular momentum is an axial vector, and its direction can be found using the right-hand screw rule.
    • Law of conservation of angular momentum: τ = dL / dt.

    Rotational Kinetic Energy

    • Rotational kinetic energy (KR) = (1/2)Iω².
    • KR = (1/2)Lω.

    Power

    • Power (P) = τω.
    • P = Iαω.
    • P = dW / dt.### Moment of Inertia and Rotational Motion
    • Moment of Inertia: a measure of an object's resistance to changes in its rotation
    • Formulae for Moment of Inertia:
      • For a disc of radius R: I = MR^2
      • For a sphere of radius R: I = (2/5)MR^2
      • For a rod of length L: I = (1/3)mL^2
    • Rotational Kinetic Energy: the energy of an object due to its rotational motion
      • Formula: KR = (1/2)Iw^2
    • Translational Kinetic Energy: the energy of an object due to its translational motion
      • Formula: KT = (1/2)mv^2

    Problems and Solutions

    • Problem 51: moment of inertia of a disc with a portion removed
      • Solution: (2/3)MR^2
    • Problem 52: moment of inertia of a circle made of a wire
      • Solution: (1/2)ml^2
    • Problem 53: moment of inertia of a semicircular ring
      • Solution: (2/3)MR^2
    • Problem 54: moment of inertia of a square made of four rods
      • Solution: (4/3)ml^2
    • Problem 55: moment of inertia of a square plate with respect to an axis passing through one corner
      • Solution: (4/3)ml^2
    • Problem 56: moment of inertia of a triangle plate
      • Solution: I1 + I2 = I3
    • Problem 57: moment of inertia of a circular disc with a small disc removed
      • Solution: (1.43)MR^2
    • Problem 58: moment of inertia of a thin uniform rod
      • Solution: (1/8)ml^2 sin^2θ
    • Problem 59: moment of inertia of a thin wire bent to form a rectangle
      • Solution: (0.4)ml^2
    • Problem 60: change in angular speed of a solid metallic sphere
      • Solution: -2%

    Angular Speed and Rotational Motion

    • Angular Speed: the rate of change of angular displacement
      • Formula: ω = Δθ/Δt
    • Problem 61: angular speed of a disc with a man walking on its edge
      • Solution: (1.2) rad/s
    • Problem 62: angular acceleration of a hollow sphere
      • Solution: (5/6)F/R
    • Problem 63: tension in a string attached to a uniform rod
      • Solution: (2)mg
    • Problem 64: angular speed gained by a ball
      • Solution: (2)v0/r

    Collision and Impulse

    • Impulse: the product of a force and the time for which it acts
      • Formula: J = FΔt
    • Problem 65: velocity of a rod after an impulse is applied
      • Solution: (1/m)J
    • Problem 66: moment of inertia of a system of two rods
      • Solution: (2/3)ml^2
    • Problem 67: moment of inertia of a thin square plate with four discs cut out
      • Solution: (3/4)MR^2
    • Problem 68: angular speed of a sphere with respect to another sphere
      • Solution: (8/3) rad/s
    • Problem 69: increase in angular speed of a turntable
      • Solution: (mvr/I0)
    • Problem 70: height of a point on a rod where a particle collides
      • Solution: (L/4)
    • Problem 71: reaction at a support A
      • Solution: (2/3)Mg

    Rotational Motion and Angular Momentum

    • Angular Momentum: a measure of an object's tendency to maintain its rotational motion
      • Formula: L = r × p
    • Problem 72: radius of a disc formed from a solid sphere
      • Solution: (7/15)R
    • Problem 73: torque required to stop a wheel
      • Solution: (2π/3) Nm
    • Problem 74: moment of inertia of a square plate about an axis making an angle with the Y-axis
      • Solution: Icos^2θ
    • Problem 75: position of the axis of rotation for minimum work
      • Solution: (1.2) m away from the 2 kg mass
    • Problem 76: moment of inertia of a square plate about an axis passing through its centre
      • Solution: I1 + I2
    • Problem 77: angular speed of a ring with two particles placed on it
      • Solution: (2/3)w
    • Problem 78: angular momentum of a ring
      • Solution: (2)MR^2w
    • Problem 79: moment of inertia of a disc about an axis passing through a point
      • Solution: (2/17)MR^2
    • Problem 80: angular speed of a cube-shaped block
      • Solution: (4/3)v/l

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    Learn about the centre of mass, its definition, and characteristics in the context of rotational motion. Understand how it relates to the study of extended objects and its position in symmetrical and asymmetrical bodies.

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