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Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
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Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
Given two vectors \( \mathbf{a} = 3\hat{i} - 4\hat{j} + 5\hat{k} \) and \( \mathbf{b} = -2\hat{i} + \hat{j} - 3\hat{k} \), the cross product \( \mathbf{a} \times \mathbf{b} \) is equal to:
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Consider the following statements regarding rotational motion about a fixed axis: (A) Only those components of external torques that lie along the axis of rotation can cause the body to rotate about the axis. (B) Torques perpendicular to the fixed axis are balanced by the constraint forces exerted by the hinges/supports. (C) Forces parallel to the axis of rotation produce non-zero torque about that axis. (D) The rate of work done by an external torque \( \tau \) rotating a body with angular speed \( \omega \) is given by \( P = \tau \omega \). How many of the above statements are correct?
Read answer →Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
In an L-shaped uniform lamina formed by three identical squares of side length \( b = 2\text{ m} \) as oriented in Fig. 6.11 (with bottom-left square at origin having vertices from \( (0,0) \) to \( (2,2) \)), the coordinate of the centre of mass is:
Read answer →Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
A particle moves with uniform speed \( v \) along a circle of radius \( R \) centered at the origin in the \( xy \)-plane. The angular momentum of the particle about a point \( P(2R, 0) \) on the \( x \)-axis varies between minimum and maximum values \( L_{\min} \) and \( L_{\max} \). The ratio \( \frac{L_{\max}}{L_{\min}} \) is:
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Consider a system of \( n \) particles where the position vector of each particle relative to the centre of mass is \( \mathbf{r}'_i \). The value of the vector sum \( \sum_{i=1}^n m_i \mathbf{r}'_i \) is always:
Read answer →Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
An engine needs to transmit a torque of \( 180 \text{ N m} \) to maintain a rotor at a uniform angular speed of \( 200 \text{ rad s}^{-1} \). If the engine operates with an efficiency of \( 75\% \), the power input supplied to the engine is:
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Given below are two statements: Statement I: The motion of the axis of a spinning top sweeping out a cone around the vertical is termed precession. Statement II: In precession of a spinning top, the point of contact of the top with the ground remains fixed. In the light of the above statements, choose the most appropriate answer from the options given below:
Read answer →Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
A projectile of mass \( 3m \) is fired from the origin with speed \( u \) at an angle of \( 45^\circ \) to the horizontal. At the highest point of its trajectory, it explodes into two fragments of mass \( m \) and \( 2m \). If the fragment of mass \( m \) immediately falls vertically downwards to the ground, at what horizontal distance from the origin does the fragment of mass \( 2m \) land? (Neglect air resistance, \( R \) is original range).
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A rigid body rotates about a fixed axis with an angular acceleration of \( 5\text{ rad/s}^2 \). If the moment of inertia of the body about this axis is \( 12\text{ kg}\cdot\text{m}^2 \), the magnitude of the external torque acting on the body is:
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Given below are two statements: Statement I: The centre of mass of a system of particles moves as if all the mass of the system was concentrated at that point and all external forces were applied there. Statement II: Internal forces between interacting particles within an isolated system can alter the total linear momentum of the system. In the light of the above statements, choose the most appropriate answer from the options given below:
Read answer →Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
An irregular lamina is suspended successively from two different points A and B. The vertical lines drawn through the points of suspension intersect at point G. This intersection point G represents the:
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Three particles of masses \( 1 \text{ kg} \), \( 2 \text{ kg} \), and \( 3 \text{ kg} \) are located at the vertices of an equilateral triangle of side length \( 2 \text{ m} \) in the \( xy \)-plane with \( 1 \text{ kg} \) at \( (0, 0) \) and \( 2 \text{ kg} \) at \( (2, 0) \). The distance of the centre of mass from the \( 1 \text{ kg} \) particle is:
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A lever of length \( 2.4\,\text{m} \) has its fulcrum placed at a distance of \( 40\,\text{cm} \) from the load end. An effort of \( 50\,\text{N} \) is applied at the other end to keep the lever in horizontal equilibrium. The load lifted by the lever and its mechanical advantage (M.A.) are, respectively:
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Three particles of masses \( 100\text{ g} \), \( 150\text{ g} \), and \( 200\text{ g} \) are placed at the vertices \( O(0,0) \), \( A(0.5\text{ m}, 0) \), and \( B(0.25\text{ m}, 0.25\sqrt{3}\text{ m}) \) of an equilateral triangle respectively. If an additional fourth mass of \( 450\text{ g} \) is placed at a point \( D \) such that the centre of mass of the four-particle system shifts to the origin \( (0,0) \), what are the coordinates of point \( D \)?
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If \( |\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = 144 \) and \( |\vec{a}| = 3 \), then the magnitude of vector \( \vec{b} \) is:
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Match List-I (Linear motion quantities) with List-II (Rotational motion analogues about a fixed axis): List-I (Linear Motion) List-II (Rotational Motion) A Mass \( M \) I Torque \( \tau \) B Force \( F \) II Moment of inertia \( I \) C Linear momentum \( p = Mv \) III Power \( P = \tau\omega \) D Power \( P = Fv \) IV Angular momentum \( L = I\omega \) Choose the correct answer from the options given below:
Read answer →Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
A light horizontal rod AB of length \( 2a \) is subjected to three parallel vertical forces: \( 2F \) downward at end A, \( F \) downward at end B, and an upward force \( 3F \) at a point C at distance \( x \) from end A. For the system to be in complete mechanical equilibrium, the value of \( x \) must be:
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A uniform square sheet of side \( 2a \) has a square of side \( a \) removed from one of its corners. Taking the corner of the original square that is diametrically opposite to the removed square as the origin, what is the distance of the centre of mass of the remaining portion from this origin?
Read answer →Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
Consider the following statements regarding the centre of mass (CM) and centre of gravity (CG) of an extended body: (A) The centre of mass of a body depends on the distribution of its mass and is independent of the gravitational field. (B) The centre of gravity of a body coincides with its centre of mass only if the gravitational field is uniform across all parts of the body. (C) In an extended non-uniform gravitational field, the net gravitational torque on the body about its centre of mass is necessarily zero. (D) The total torque on a system is independent of the choice of origin if the net external force on the system is zero. Choose the correct answer from the options given below:
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A projectile is moving along a parabolic trajectory. In mid-air, it explodes into several fragments due to an internal explosion. Neglecting air resistance, the centre of mass of the fragments:
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An ideal rigid body is defined as a system of particles in which the distances between all pairs of particles do not change under the influence of forces. Which of the following best describes the real-world validity of this model?
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Four particles of masses \( 1\text{ kg} \), \( 2\text{ kg} \), \( 3\text{ kg} \), and \( 4\text{ kg} \) have position vectors \( \mathbf{r}_1 = (\hat{\mathbf{i}} + 2\hat{\mathbf{j}} + 3\hat{\mathbf{k}})\text{ m} \), \( \mathbf{r}_2 = (2\hat{\mathbf{i}} - \hat{\mathbf{j}} + \hat{\mathbf{k}})\text{ m} \), \( \mathbf{r}_3 = (-\hat{\mathbf{i}} + 3\hat{\mathbf{j}} - 2\hat{\mathbf{k}})\text{ m} \), and \( \mathbf{r}_4 = (3\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 2\hat{\mathbf{k}})\text{ m} \). The magnitude of the position vector of their centre of mass is:
Read answer →Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
