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Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
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Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
A child stands at the centre of a turntable with outstretched arms rotating with an initial rotational kinetic energy \( K_0 \). When the child folds their arms, the moment of inertia of the system reduces to \( \frac{2}{5} \) of its initial value. Assuming no external torque acts, the ratio of the new rotational kinetic energy to the initial rotational kinetic energy (\( K_f / K_0 \)) is:
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A wheel rotating about a fixed axis with an initial angular speed of \( 20\text{ rad/s} \) is decelerated uniformly at \( 2\text{ rad/s}^2 \). The ratio of the tangential acceleration to the centripetal acceleration of a point on the rim at \( t = 5\text{ s} \) (radius of the wheel = \( 0.2\text{ m} \)) is:
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Match List-I with List-II: List-I (Physical Quantity) List-II (Defining Vector Relation) A Linear velocity in rotation I \( \frac{d\vec{\omega}}{dt} \) B Torque (Moment of force) II \( \vec{\omega} \times \vec{r} \) C Newton's second law for system III \( \vec{r} \times \vec{F} \) D Angular acceleration vector IV \( \frac{d\vec{P}}{dt} = \vec{F}_{\text{ext}} \) Choose the correct answer from the options given below:
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A car weighs \( 1800 \text{ kg} \). The distance between its front and back axles is \( 1.80 \text{ m} \). Its centre of gravity is \( 1.05 \text{ m} \) behind the front axle. Assuming \( g = 10 \text{ m s}^{-2} \), the normal force exerted by the level ground on each front wheel is:
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Refer to the given figure showing an equilateral triangular lamina divided into strips parallel to the base \( MN \). The centre of mass of each individual narrow strip lies at its midpoint, so the centre of mass of the whole lamina lies on the median \( LP \). By similarly considering strips parallel to the other two sides, the centre of mass of the entire triangular lamina \( LMN \) must be located at:
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Consider the following bodies, each of mass \( M \) and radius \( R \): (A) Thin circular ring about its central perpendicular axis (B) Circular disc about its diameter (C) Solid sphere about its diameter (D) Solid cylinder about its geometric axis Which of the following represents the correct descending order of their moments of inertia? Which of the above statements is/are correct?
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The angular momentum of a system of particles varies with time as \( \mathbf{L}(t) = (4t^2\hat{i} - 6t\hat{j} + 2\hat{k})\,\text{kg}\cdot\text{m}^2/\text{s} \). What is the magnitude of the net external torque acting on the system at \( t = 1\,\text{s} \)?
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Given below are two statements: Statement I: In pure translational motion, all particles of a rigid body have the same velocity at any given instant. Statement II: When a cylinder rolls without slipping down an inclined plane, the point of contact with the plane has a non-zero velocity directed down the incline. In the light of the above statements, choose the most appropriate answer from the options given below:
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A flywheel has moment of inertia \( I = 0.8 \text{ kg m}^2 \). A steady torque accelerates it from an initial angular speed of \( 10 \text{ rad s}^{-1} \) to \( 30 \text{ rad s}^{-1} \) in \( 4 \text{ s} \). The instantaneous power delivered by the torque at \( t = 2 \text{ s} \) is:
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A particle of mass \( m = 0.5 \text{ kg} \) rotates in the \( xy \)-plane in a circle of radius \( r = 2 \text{ m} \) centred at the origin with constant angular speed \( \omega = 4 \text{ rad s}^{-1} \). If the origin is shifted along the \( z \)-axis such that the position vector is \( \mathbf{r} = \mathbf{r}_{\text{in-plane}} + 3\mathbf{\hat{k}} \), the magnitude of the component of angular momentum perpendicular to the axis of rotation (\( |\mathbf{L}_{\perp}| \)) is:
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In a binary star system, two stars of masses \( m_1 \) and \( m_2 \) separated by distance \( d \) rotate in circular orbits around their common centre of mass under their mutual gravitational attraction. If \( r_1 \) and \( r_2 \) are the radii of their orbits and \( v_1 \) and \( v_2 \) are their respective orbital speeds, which of the following relations is correct?
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Referring to the ladder setup shown in Figure, a ladder of length 3 m leans against a frictionless wall with foot at \( A \) and top at \( B \). The normal reaction of the wall is \( F_1 \) and the floor reaction components are normal force \( N \) and friction \( F \). For rotational equilibrium about point \( A \), the equation relating \( F_1 \) and the weight \( W \) of the ladder is:
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In the system shown in Figure, a uniform bar AB of length 70 cm and weight \( W \) is supported on knife edges \( K_1 \) and \( K_2 \) located at 10 cm from ends A and B respectively. A load \( W_1 \) is suspended at P, 30 cm from end A. If the reactions at \( K_1 \) and \( K_2 \) are denoted as \( R_1 \) and \( R_2 \), taking the moments about the centre of gravity G of the bar yields which of the following equilibrium conditions?
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Given below are two statements: Statement I: The centre of gravity of a small, homogeneous body coincides with its centre of mass. Statement II: The centre of gravity of an irregular flat cardboard can be determined experimentally by suspending it freely from two or more points and finding the intersection of the plumb lines. In the light of the above statements, choose the most appropriate answer from the options given below:
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A constant torque of \( 25\text{ N}\cdot\text{m} \) acts on a body rotating about a fixed axis with moment of inertia \( 2.5\text{ kg}\cdot\text{m}^2 \). If the body starts from rest, the work done by the torque in the first 4 seconds is:
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Refer to the given figure showing a rigid body rotating about the fixed \( z \)-axis. If particle \( P_1 \) moves in a circle of radius \( r_1 \) in plane \( C_1 \) and particle \( P_2 \) moves in a circle of radius \( r_2 \) in plane \( C_2 \) (\( r_2 > r_1 \)), with particle \( P_3 \) lying directly on the rotation axis, which of the following relations is correct regarding their linear speeds \( v_1, v_2, v_3 \) and angular velocities \( \omega_1, \omega_2, \omega_3 \)?
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The position vector of a particle in a rotating rigid body is given by \( \vec{r} = (3\hat{i} + 4\hat{j} - 2\hat{k})\text{ m} \) with respect to the origin on the axis of rotation. If the angular velocity of the rigid body is \( \vec{\omega} = (2\hat{i} - 3\hat{j} + \hat{k})\text{ rad/s} \), the linear velocity of the particle is:
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A light rigid rod of length \( l \) has two point masses, each of mass \( \frac{M}{2} \), attached at its two ends. The moment of inertia of this system about an axis passing through its centre and perpendicular to the rod is:
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Consider the following statements regarding levers and the centre of gravity: (A) The Mechanical Advantage (M.A.) of a lever is defined as the ratio of load to effort (\( F_1/F_2 \)). (B) The Mechanical Advantage of an ideal lever is mathematically given by \( d_1/d_2 \), where \( d_1 \) is the load arm and \( d_2 \) is the effort arm. (C) The moment of a couple acting on a rigid body varies with the choice of the reference point about which moments are computed. (D) In a uniform gravitational field, the centre of gravity of a body coincides precisely with its centre of mass. Choose the correct answer from the options given below:
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A solid cylinder of mass \( 20 \text{ kg} \) and radius \( 0.25 \text{ m} \) rotates about its axis of symmetry with angular speed \( 100 \text{ rad s}^{-1} \). The magnitude of the angular momentum of the cylinder about its axis and the kinetic energy associated with rotation are respectively:
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Two stars of equal mass \( M \) revolve in circular orbits of radius \( R \) about their common centre of mass under mutual gravitational attraction. If the orbital speed of each star is \( v \), what would be the new orbital speed of each star if the separation between them is halved, keeping their circular orbits about the centre of mass?
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A rigid body rotates about a fixed axis with an angular speed \( \omega \). A particle \( P_1 \) is at a perpendicular distance \( r_1 = 0.2\text{ m} \) from the axis, and another particle \( P_2 \) is at a perpendicular distance \( r_2 = 0.6\text{ m} \) from the axis. If the linear velocity of \( P_1 \) is \( 4\text{ m/s} \), the centripetal acceleration of \( P_2 \) is:
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