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Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
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Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
A force \( \mathbf{F} = (2\hat{i} - 4\hat{j} + 5\hat{k})\,\text{N} \) acts at the point \( A(1, 2, -1)\,\text{m} \). The torque of this force about a point \( B(2, -1, 3)\,\text{m} \) is:
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For a spinning top whose tip is fixed on the ground at point \( O \), the movement of its axis of rotation sweeping out a cone around the vertical is termed:
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Consider the following statements regarding the mechanical equilibrium of a rigid body: (A) A rigid body is in complete mechanical equilibrium if both its total linear momentum and total angular momentum do not change with time. (B) If the net external force acting on a rigid body is zero, the total torque on the body is independent of the choice of origin. (C) A couple comprises two equal and opposite forces acting along two distinct, parallel lines of action. (D) A rigid body acted upon solely by an unbalanced couple cannot be in rotational equilibrium. Which of the above statements are correct?
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A non-uniform bar of weight \( W \) and length \( L = 2.0 \text{ m} \) is suspended horizontally at rest by two light strings attached to its ends. The strings make angles \( 36.9^\circ \) and \( 53.1^\circ \) with the vertical respectively (given \( \tan 36.9^\circ = 0.75 \), \( \tan 53.1^\circ = 1.33 \)). The distance \( d \) of the centre of gravity of the bar from its left end is approximately:
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A solid cylinder and a hollow cylinder have the same mass \( M \) and same outer radius \( R \). They rotate about their respective central longitudinal axes with angular speeds \( \omega_1 \) and \( \omega_2 \). If both cylinders possess the same rotational kinetic energy, the ratio of their angular speeds \( \omega_1 / \omega_2 \) is:
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A thin uniform rod of length \( L \) lies along the \( x \)-axis from \( x = 0 \) to \( x = L \). It is placed in a non-uniform gravitational field where acceleration due to gravity is directed along \( -\hat{j} \) and has magnitude \( g(x) = g_0(1 + \alpha x) \). The \( x \)-coordinate of the centre of gravity of the rod is:
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Given below are two statements: Statement I: The moment of a force vanishes if the directional line of action of the force passes through the chosen origin. Statement II: If the directions of both the position vector \( \mathbf{r} \) and the force vector \( \mathbf{F} \) are reversed, the direction of the torque vector \( \boldsymbol{\tau} \) remains unchanged. In the light of the above statements, choose the most appropriate answer from the options given below:
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A thin uniform circular ring of mass \( M \) and radius \( R \) has a moment of inertia \( I_0 \) about an axis perpendicular to its plane and passing through its centre. If the ring is cut into two equal semicircular halves, the moment of inertia of one semicircular half of mass \( M/2 \) about the same original central axis is:
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Given below are two statements: Statement I: The SI unit of radius of gyration is \( \text{kg}\cdot\text{m} \). Statement II: The dimensional formula of moment of inertia is \( [M L^2 T^{-1}] \). In the light of the above statements, choose the most appropriate answer from the options given below:
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Two particles of masses \( m_1 = 2\text{ kg} \) and \( m_2 = 6\text{ kg} \) are located on the \( x \)-axis at \( x_1 = 1\text{ m} \) and \( x_2 = 5\text{ m} \) respectively. If the mass \( m_1 \) is shifted towards the right by \( 3\text{ m} \), by what distance must \( m_2 \) be shifted so that the centre of mass of the system shifts to the right by \( 1.5\text{ m} \)?
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A ladder of length \( 3\text{ m} \) and mass \( 20\text{ kg} \) leans against a smooth frictionless vertical wall with its foot placed \( 1\text{ m} \) from the wall. Assuming the center of gravity is at its midpoint, the normal reaction force exerted by the vertical wall on the ladder is: (Take \( g = 9.8\text{ m/s}^2 \))
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Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): Torque and work possess identical dimensional formulas \( [M L^2 T^{-2}] \), yet work is a scalar quantity whereas torque is a vector quantity. Reason (R): Torque is defined via the vector cross product \( \mathbf{r} \times \mathbf{F} \), giving it an explicit directional character, whereas work is defined via the scalar dot product \( \mathbf{F} \cdot \Delta\mathbf{r} \). In the light of the above statements, choose the correct answer from the options given below:
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For a system of \( n \) particles, if the total external force acting on the system is zero, which of the following physical quantities must remain constant?
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Three particles of masses \( m \), \( 2m \), and \( 3m \) are placed on the \( x \)-axis at positions \( x = 0 \), \( x = L \), and \( x = 2L \) respectively. A fourth particle of mass \( 4m \) is added at position \( x = x_4 \) such that the centre of mass of the four-particle system shifts to \( x = L \). The coordinate \( x_4 \) must be:
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A stationary radium nucleus (\( ^{226}\text{Ra} \)) undergoes alpha decay into a radon nucleus (\( ^{222}\text{Rn} \)) and an alpha particle (\( ^{4}\text{He} \)). If the total kinetic energy released in the decay is \( Q \), what fraction of the total kinetic energy is carried away by the radon nucleus?
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For a thin uniform rod of length \( L \) and mass \( M \) lying along the \( x \)-axis with its geometric centre at the origin \( x = 0 \), the integral \( \int x\,dm \) evaluated over the rod is:
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A rigid system consists of a light rod of length \( l \) with two masses \( M/2 \) attached at each end. If the radius of gyration of this system about an axis perpendicular to the rod passing through its centre of mass is \( k_1 \), and the radius of gyration of a thin uniform ring of radius \( R \) about its diameter is \( k_2 \), then the ratio \( k_1 / k_2 \) when \( l = 2R \) is:
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Consider the following statements regarding the analogues between translational motion and rotational motion about a fixed axis: (A) Angular momentum \( L = I \omega \) is analogous to linear momentum \( p = M v \). (B) Torque \( \tau = I \alpha \) is analogous to force \( F = M a \). (C) Rotational work \( W = \tau \theta \) is analogous to translational work \( W = F s \). (D) Moment of inertia is analogous to acceleration. How many of the above statements are correct?
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A rigid body rotates about a fixed axis with angular velocity \( \vec{\omega} = \omega\hat{k} \). If the position vector of a point \( P \) on the body from a point \( O \) on the rotation axis is \( \vec{r} = 3\hat{i} + 4\hat{j} + 5\hat{k} \), the ratio of the linear speed of point \( P \) to the product \( \omega |\vec{r}| \) is:
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Three vectors define the edges of a parallelepiped: \( \mathbf{a} = 2\mathbf{\hat{i}} + 3\mathbf{\hat{j}} - \mathbf{\hat{k}} \), \( \mathbf{b} = \mathbf{\hat{i}} - 2\mathbf{\hat{j}} + 2\mathbf{\hat{k}} \), and \( \mathbf{c} = 3\mathbf{\hat{i}} - \mathbf{\hat{j}} + \lambda\mathbf{\hat{k}} \). If the volume of the parallelepiped formed by these vectors is \( 45 \text{ cubic units} \), one of the possible values of \( \lambda \) is:
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In an \( \text{HCl} \) molecule, the separation between the nuclei of the hydrogen and chlorine atoms is \( 1.27 \text{ \AA} \). If a chlorine atom is \( 35.5 \) times as massive as a hydrogen atom, the location of the centre of mass of the molecule measured from the hydrogen nucleus is:
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The physical quantity in rotational motion that plays the same role as mass does in translational motion is:
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