Chapter 5

The centre of mass of a systems of two particles is (A) on the line joining them and midway between them (B) on the line joining them at a point whose distance from each particle is proportional to the square of the mass of that particle. (C) on the line joining them at a point whose distance from each particle inversely proportional to the mass of that particle. (D) On the line joining them at a point whose distance from each particle is proportional to the mass of that particle.

Particles of \(1 \mathrm{gm}, 1 \mathrm{gm}, 2 \mathrm{gm}, 2 \mathrm{gm}\) are placed at the corners \(A, B, C, D\), respectively of a square of side \(6 \mathrm{~cm}\) as shown in figure. Find the distance of centre of mass of the system from geometrical centre of square. (A) \(1 \mathrm{~cm}\) (B) \(2 \mathrm{~cm}\) (C) \(3 \mathrm{~cm}\) (D) \(4 \mathrm{~cm}\)

Three particles of the same mass lie in the \((\mathrm{X}, \mathrm{Y})\) plane, The \((X, Y)\) coordinates of their positions are \((1,1),(2,2)\) and \((3,3)\) respectively. The \((X, Y)\) coordinates of the centre of mass are \(\\{\mathrm{A}\\}(1,2)\) \(\\{\mathrm{B}\\}(2,2)\) \(\\{\mathrm{C}\\}(1.5,2)\) \(\\{\mathrm{D}\\}(2,1.5)\)

Consider a two-particle system with the particles having masses \(\mathrm{M}_{1}\), and \(\mathrm{M}_{2}\). If the first particle is pushed towards the centre of mass through a distance \(d\), by what distance should the second particle be moved so as to keep the centre of mass at the same position? \(\\{\mathrm{A}\\}\left[\left(\mathrm{M}_{1} \mathrm{~d}\right) /\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right)\right]\) \(\\{\mathrm{B}\\}\left[\left(\mathrm{M}_{2} \mathrm{~d}\right) /\left(\mathrm{M}_{1}+\mathrm{M}_{2}\right)\right]\) \(\\{\mathrm{C}\\}\left[\left(\mathrm{M}_{1} \mathrm{~d}\right) /\left(\mathrm{M}_{2}\right)\right]\) \(\\{\mathrm{D}\\}\left[\left(\mathrm{M}_{2} \mathrm{~d}\right) /\left(\mathrm{M}_{1}\right)\right]\)

From a uniform circular disc of radius \(\mathrm{R}\), a circular disc of radius \(\mathrm{R} / 6\) and having centre at a distance \(+\mathrm{R} / 2\) from the centre of the disc is removed. Determine the centre of mass of remaining portion of the disc. \(\\{\mathrm{A}\\}[(-\mathrm{R}) / 70]\) \(\\{\mathrm{B}\\}[(+\mathrm{R}) / 70]\) \(\\{\mathrm{C}\\}[(-\mathrm{R}) / 7]\) \(\\{\mathrm{D}\\}[(+\mathrm{R}) / 7]\)

A circular plate of uniform thickness has a diameter of \(56 \mathrm{~cm}\). A circular portion of diameter \(42 \mathrm{~cm} .\) is removed from tve \(\mathrm{x}\) edge of the plate. Find the position of centre of mass of the remaining portion with respect to centre of mass of whole plate. \(\\{\mathrm{A}\\}-7 \mathrm{~cm} \quad\\{\mathrm{~B}\\}+9 \mathrm{~cm} \quad\\{\mathrm{C}\\}-9 \mathrm{~cm} \quad\\{\mathrm{D}\\}+7 \mathrm{~cm}\)

Two blocks of masses \(10 \mathrm{~kg}\) an \(4 \mathrm{~kg}\) are connected by a spring of negligible mass and placed on a frictionless horizontal surface. An impulse gives velocity of \(14 \mathrm{~m} / \mathrm{s}\) to the heavier block in the direction of the lighter block. The velocity of the centre of mass is : \(\\{\mathrm{A}\\} 30 \mathrm{~m} / \mathrm{s}\) \(\\{\mathrm{B}\\} 20 \mathrm{~m} / \mathrm{s}\) \(\\{\mathrm{C}\\} 10 \mathrm{~m} / \mathrm{s}\) \(\\{\mathrm{D}\\} 5 \mathrm{~m} / \mathrm{s}\)

A particle performing uniform circular motion has angular momentum \(L\)., its angular frequency is doubled and its \(K . E\). halved, then the new angular momentum is \(\\{\mathrm{A}\\} 1 / 2\) \\{B \(\\} 1 / 4\) \(\\{\mathrm{C}\\} 2 \mathrm{~L}\) \(\\{\mathrm{D}\\} 4 \mathrm{~L}\)

A circular disc of radius \(\mathrm{R}\) is removed from a bigger disc of radius \(2 \mathrm{R}\). such that the circumferences of the disc coincide. The centre of mass of the remaining portion is \(\alpha R\) from the centre of mass of the bigger disc. The value of \(\alpha\) is. \(\\{\mathrm{A}\\} 1 / 2\) \\{B \\} \(1 / 6\) \\{C\\} \(1 / 4\) \(\\{\mathrm{D}\\}[(-1) / 3]\)

A body of mass \(\mathrm{m}\) is tied to one end of spring and whirled round in a horizontal plane with a constant angular velocity. The elongation in the spring is one centimetre. If the angular velocity is doubted, the elongation in the spring is \(5 \mathrm{~cm}\). The original length of spring is... \(\\{\mathrm{A}\\} 16 \mathrm{~cm}\) \(\\{B\\} 15 \mathrm{~cm}\) \(\\{\mathrm{C}\\} 14 \mathrm{~cm}\) \(\\{\mathrm{D}\\} 13 \mathrm{~cm}\)

A cylinder of mass \(5 \mathrm{~kg}\) and radius \(30 \mathrm{~cm}\), and free to rotate about its axis, receives an angular impulse of \(3 \mathrm{~kg} \mathrm{M}^{2} \mathrm{~S}^{-1}\) initially followed by a similar impulse after every \(4 \mathrm{sec}\). what is the angular speed of the cylinder 30 sec after initial impulse? The cylinder is at rest initially. \(\\{\mathrm{A}\\} 106.7 \mathrm{rad} \mathrm{S}^{-1}\) \\{B\\} \(206.7 \mathrm{rad} \mathrm{S}^{-1}\) \\{C\\} \(107.6 \mathrm{rad} \mathrm{S}^{-1}\) \\{D \(\\} 207.6 \mathrm{rad} \mathrm{S}^{-1}\)

Two circular loop \(A \& B\) of radius \(r_{A}\) and \(r_{B}\) respectively are made from a uniform wire. The ratio of their moment of inertia about axes passing through their centres and perpendicular to their planes is \(\left(\mathrm{I}_{\mathrm{B}} / \mathrm{I}_{\mathrm{A}}\right)=8\) then \(\left(\mathrm{r}_{\mathrm{B}} / \mathrm{r}_{\mathrm{A}}\right)\) Ra is equal to... \(\\{\mathrm{A}\\} 2\) \(\\{B\\} 4\) \\{C \(\\}\) \(\\{\mathrm{D}\\} 8\)

If the earth were to suddenly contract so that its radius become half of it present radius, without any change in its mass, the duration of the new day will be... \(\\{\mathrm{A}\\} 6 \mathrm{hr}\) \\{B \(12 \mathrm{hr}\) \(\\{\mathrm{C}\\} 18 \mathrm{hr}\) \\{D \(\\} 30 \mathrm{hr}\)

In HCl molecule the separation between the nuclei of the two atoms is about \(1.27 \mathrm{~A}\left(1 \mathrm{~A}=10^{-10}\right)\). The approximate location of the centre of mass of the molecule is \(-\mathrm{A}\) i \(\wedge\) with respect of Hydrogen atom (mass of CL is \(35.5\) times of mass of Hydrogen \()\) \(\\{\mathrm{A}\\} 1 \mathrm{i}\) \\{B \\} \(2.5 \mathrm{i}\) \\{C \(\\} 1.24 \mathrm{i}\) \\{D \(1.5 \mathrm{i}\)

Two bodies of mass \(1 \mathrm{~kg}\) and \(3 \mathrm{~kg}\) have position vector \((\mathrm{i} \wedge+2 \mathrm{j} \wedge+\mathrm{k} \wedge)\) and \((-3 \mathrm{i}-2 \mathrm{j}+\mathrm{k})\) respectively the centre of mass of this system has a position vector ..... \(\\{\mathrm{A}\\}-2 \mathrm{i} \wedge+2 \mathrm{k} \wedge\) \(\\{B\\}-2 i \wedge-j \wedge+k \wedge\) \(\\{C\\} 2 i \wedge-j \wedge-k \wedge\) \(\\{\mathrm{D}\\}-\mathrm{i} \wedge+\mathrm{j} \wedge+\mathrm{k} \wedge\)

Identify the correct statement for the rotational motion of a rigid body \(\\{A\\}\) Individual particles of the body do not undergo accelerated motion \\{B \\} The centre of mass of the body remains unchanged. \\{C\\} The centre of mass of the body moves uniformly in a circular path \\{D\\} Individual particle and centre of mass of the body undergo an accelerated motion.

A car is moving at a speed of \(72 \mathrm{~km} / \mathrm{hr}\) the radius of its wheel is \(0.25 \mathrm{~m}\). If the wheels are stopped in 20 rotations after applying breaks then angular retardation produced by the breaks is \(\ldots .\) \(\\{\mathrm{A}\\}-25.5 \mathrm{rad} / \mathrm{s}^{2}\) \(\\{\mathrm{B}\\}-29.52 \mathrm{rad} / \mathrm{s}^{2}\) \(\\{\mathrm{C}\\}-33.52 \mathrm{rad} / \mathrm{s}^{2}\) \(\\{\mathrm{D}\\}-45.52 \mathrm{rad} / \mathrm{s}^{2}\)

A wheel rotates with a constant acceleration of \(2.0 \mathrm{rad} / \mathrm{sec}^{2}\) If the wheel start from rest. The number of revolution it makes in the first ten seconds will be approximately. \(\\{\mathrm{A}\\} 8\) \\{B \\} 16 \(\\{\mathrm{C}\\} 24\) \(\\{\mathrm{D}\\} 32\)

Two discs of the same material and thickness have radii \(0.2 \mathrm{~m}\) and \(0.6 \mathrm{~m}\) their moment of inertia about their axes will be in the ratio \(\\{\mathrm{A}\\} 1: 81\) \(\\{\mathrm{B}\\} 1: 27\) \(\\{C\\} 1: 9\) \(\\{\mathrm{D}\\} 1: 3\)

A wheel of mass \(10 \mathrm{~kg}\) has a moment of inertia of \(160 \mathrm{~kg} \mathrm{~m}\) radius of gyration will be \(\begin{array}{llll}\\{\mathrm{A}\\} 10 & \\{\mathrm{~B}\\} 8 & \\{\mathrm{C}\\} 6 & \\{\mathrm{D}\\} 4\end{array}\)

One circular rig and one circular disc both are having the same mass and radius. The ratio of their moment of inertia about the axes passing through their centers and perpendicular to their planes, will be \(\\{\mathrm{A}\\} 1: 1\) \(\\{\mathrm{B}\\} 2: 1\) \(\\{C\\} 1: 2\) \(\\{\mathrm{D}\\} 4: 1\)

One solid sphere \(\mathrm{A}\) and another hollow sphere \(\mathrm{B}\) are of the same mass and same outer radii. The moment of inertia about their diameters are respectively \(\mathrm{I}_{\mathrm{A}}\) and \(\mathrm{I}_{\mathrm{B}}\) such that... \(\\{\mathrm{A}\\} \mathrm{I}_{\mathrm{A}}=\mathrm{I}_{\mathrm{B}}\) \(\\{\mathrm{B}\\} \mathrm{I}_{\mathrm{A}}>\mathrm{I}_{\mathrm{B}}\) \(\\{\mathrm{C}\\} \mathrm{I}_{\mathrm{A}}<\mathrm{I}_{\mathrm{B}}\) \(\\{\mathrm{D}\\}\left(\mathrm{I}_{\mathrm{A}} / \mathrm{I}_{\mathrm{B}}\right)=(\mathrm{d} \mathrm{A} / \mathrm{dB})\) (radio of their densities)

A ring of mass \(\mathrm{M}\) and radius \(\mathrm{r}\) is melted and then molded in to a sphere then the moment of inertia of the sphere will be..... \(\\{\mathrm{A}\\}\) more than that of the ring \\{B \\} Less than that of the ring \(\\{\mathrm{C}\\}\) Equal to that of the ring \\{D\\} None of these

A circular disc of radius \(\mathrm{R}\) and thickness \(\mathrm{R} / 6\) has moment of inertia I about an axis passing through its centre and perpendicular to its plane. It is melted and re-casted in to a solid sphere. The moment of inertia of the sphere about its diameter as axis of rotation is \(\ldots\) \(\\{\mathrm{A}\\} \mathrm{I}\) \(\\{\mathrm{B}\\}(2 \mathrm{I} / 8)\) \(\\{\mathrm{C}\\}(\mathrm{I} / 5)\) \(\\{\mathrm{D}\\}(\mathrm{I} / 10)\)

One quarter sector is cut from a uniform circular disc of radius \(\mathrm{R}\). This sector has mass \(\mathrm{M}\). It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation is... \(\\{\mathrm{A}\\}(1 / 2) \mathrm{MR}^{2}\) \(\\{\mathrm{B}\\}(1 / 4) \mathrm{MR}^{2}\) \(\\{\mathrm{C}\\}(1 / 8) \mathrm{MR}^{2}\) \\{D \(\\} \sqrt{2} \mathrm{MR}^{2}\)

Two disc of same thickness but of different radii are made of two different materials such that their masses are same. The densities of the materials are in the ratio \(1: 3\). The moment of inertia of these disc about the respective axes passing through their centres and perpendicular to their planes will be in the ratio. \(\\{\mathrm{A}\\} 1: 3\) \\{B\\} \(3: 1\) \\{C\\} \(1: 9\) \(\\{\mathrm{D}\\} 9: 1\)

Let I be the moment of inertia of a uniform square plate about an axis \(\mathrm{AB}\) that passes through its centre and is parallel to two of its sides \(\mathrm{CD}\) is a line in the plane of the plate that passes through the centre of the plate and makes an angle of \(\theta\) with \(\mathrm{AB}\). The moment of inertia of the plate about the axis \(\mathrm{CD}\) is then equal to.... \(\\{\mathrm{A}\\} \mathrm{I}\) \(\\{B\\} I \sin ^{2} \theta\) \(\\{C\\} I \cos ^{2} \theta\) \(\\{\mathrm{D}\\} I \cos ^{2}(\theta / 2)\)

A small disc of radius \(2 \mathrm{~cm}\) is cut from a disc of radius $6 \mathrm{~cm}\(. If the distance between their centres is \)3.2 \mathrm{~cm}$, what is the shift in the centre of mass of the disc... $\begin{array}{llll}\\{\mathrm{A}\\}-0.4 \mathrm{~cm} & \\{\mathrm{~B}\\}-2.4 \mathrm{~cm} & \\{\mathrm{C}\\}-1.8 \mathrm{~cm} & \\{\mathrm{D}\\} & 1.2 \mathrm{~cm}\end{array}$

A straight rod of length \(L\) has one of its ends at the origin and the other end at \(\mathrm{x}=\mathrm{L}\) If the mass per unit length of rod is given by Ax where \(A\) is constant where is its centre of mass. \(\\{\mathrm{A}\\} \mathrm{L} / 3\) \(\\{\mathrm{B}\\} \mathrm{L} / 2\) \(\\{\mathrm{C}\\} 2 \mathrm{~L} / 3\) \(\\{\mathrm{D}\\} 3 \mathrm{~L} / 4\)

A thin circular ring of mass \(\mathrm{M}\) and radius \(\mathrm{r}\) is rotating about its axis with a constant angular velocity \(\mathrm{w}\). Two objects each of mass \(\mathrm{m}\) are attached gently to the opposite ends of a diameter of the ring. The ring will now rotate with an angular velocity.... \(\\{\mathrm{A}\\}[\\{\omega(\mathrm{M}-2 \mathrm{~m})\\} /\\{\mathrm{M}+2 \mathrm{~m}\\}]\) \(\\{\mathrm{B}\\}[\\{\omega \mathrm{M}\\} /\\{\mathrm{M}+2 \mathrm{~m}\\}]\) \(\\{C\\}[\\{\omega M)\\} /\\{M+m\\}]\) \(\\{\mathrm{D}\\}[\\{\omega(\mathrm{M}+2 \mathrm{~m})\\} / \mathrm{M}]\)

Two point masses of \(0.3 \mathrm{~kg}\) and \(0.7 \mathrm{~kg}\) are fixed at the ends of a rod of length \(1.4 \mathrm{~m}\) and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum, is located at a distance of ..... \(\\{\mathrm{A}\\} 0.4 \mathrm{~m}\) from mass of \(0.3 \mathrm{~kg}\) \\{B \(0.98 \mathrm{~m}\) from mass of \(0.3 \mathrm{~kg}\) \\{C\\} \(0.7 \mathrm{~m}\) from mass of \(0.7 \mathrm{~kg}\) \\{D \(\\} 0.98 \mathrm{~m}\) from mass of \(0.7 \mathrm{~kg}\)

In a bicycle the radius of rear wheel is twice the radius of front wheel. If \(\mathrm{r}_{\mathrm{F}}\) and \(\mathrm{r}_{\mathrm{r}}\) are the radius, \(\mathrm{v}_{\mathrm{F}}\) and \(\mathrm{v}_{\mathrm{r}}\) are speed of top most points of wheel respectively then... \(\\{\mathrm{A}\\} \mathrm{v}_{\mathrm{r}}=2 \mathrm{v}_{\mathrm{F}}\) \(\\{\mathrm{B}\\} \mathrm{v}_{\mathrm{F}}=2 \mathrm{v}_{\mathrm{r}} \quad\\{\mathrm{C}\\} \mathrm{v}_{\mathrm{F}}=\mathrm{v}_{\mathrm{r}}\) \(\\{\mathrm{D}\\} \mathrm{v}_{\mathrm{F}}>\mathrm{v}_{\mathrm{r}}\)

From a circular disc of radius \(\mathrm{R}\) and mass \(9 \mathrm{M}\), a small disc of radius \(\mathrm{R} / 3\) is removed from the disc. The moment of inertia of the remaining portion about an axis perpendicular to the plane of the disc and passing through \(\mathrm{O}\) is.... \(\\{\mathrm{A}\\} 4 \mathrm{MR}^{2}\) \(\\{\mathrm{B}\\}(40 / 9) \mathrm{MR}^{2}\) \(\\{\mathrm{C}\\} 10 \mathrm{MR}^{2}\) \(\\{\mathrm{D}\\}(37 / 9) \mathrm{MR}^{2}\)

If the earth is treated as a sphere of radius \(R\) and mass \(M\). Its angular momentum about the axis of rotation with period \(\mathrm{T}\) is..... \(\\{\mathrm{A}\\}\left(\pi \mathrm{MR}^{3} / \mathrm{T}\right)\) \(\\{\mathrm{B}\\}\left(\operatorname{MR}^{2} \pi / \mathrm{T}\right)\) \(\\{C\\}\left(2 \pi \mathrm{MR}^{2} / 5 \mathrm{~T}\right)\) \(\\{\mathrm{D}\\}\left(4 \pi \mathrm{MR}^{2} / 5 \mathrm{~T}\right)\)

The M.I. of a body about the given axis is \(1.2 \mathrm{kgm}^{2}\) initially the body is at rest. In order to produce a rotational kinetic energy of \(1500 \mathrm{~J}\). an angular acceleration of \(25 \mathrm{rad} \mathrm{sec}^{2}\) must be applied about that axis for duration of \(\ldots\) \(\\{\mathrm{A}\\} 4 \mathrm{sec}\) \\{B \(\\} 2 \mathrm{sec}\) \(\\{C\\} 8 \mathrm{sec}\) \(\\{\mathrm{D}\\} 10 \mathrm{sec}\)

An automobile engine develops \(100 \mathrm{kw}\) when rotating at a speed of 1800 r.p.m. what torque does it deliver? \(\\{\mathrm{A}\\} 350 \mathrm{Nm} \quad\\{\mathrm{B}\\} 440 \mathrm{Nm}\) \\{C \(\\} 531 \mathrm{Nm} \quad\\{\mathrm{D}\\} 628 \mathrm{Nm}\)

If distance of the earth becomes three times that of the present distance from the sun then number of days in one year will be .... \(\\{\mathrm{A}\\}[365 \times 3]\) \(\\{\mathrm{B}\\}[365 \times 27]\) \(\\{\mathrm{C}\\}[365 \times(3 \sqrt{3})]\) \(\\{\mathrm{D}\\}[365 /(3 \sqrt{3})]\)

A solid sphere and a solid cylinder having same mass and radius roll down the same incline the ratio of their acceleration will be.... \(\\{\mathrm{A}\\} 15: 14\) \(\\{\mathrm{B}\\} 14: 15\) \(\\{\mathrm{C}\\} 5: 3\) \(\\{\mathrm{D}\\} 3: 5\)

The ratio of angular momentum of the electron in the first allowed orbit to that in the second allowed orbit of hydrogen atom is ...... \(\\{\mathrm{A}\\} \sqrt{2}\) \(\\{B\\} \sqrt{(1 / 2)}\) \(\\{\mathrm{C}\\}(1 / 2) \quad\\{\mathrm{D}\\} 2\)

A player caught a cricket ball of mass \(150 \mathrm{gm}\) moving at a rate of \(20 \mathrm{~m} / \mathrm{s}\) If the catching process is Completed in \(0.1\) sec the force of the flow exerted by the ball on the hand of the player ..... N \(\\{\mathrm{A}\\} 3\) \(\\{B\\} 30\) \(\\{\mathrm{C}\\} 150\) \(\\{\mathrm{D}\\} 300\)

Two identical hollow spheres of mass \(\mathrm{M}\) and radius \(\mathrm{R}\) are joined together and the combination is rotated about an axis tangential to one sphere and perpendicular to the line connecting their centres. The moment of inertia of the combination is \(\\{\mathrm{A}\\} 10 \mathrm{MR}^{2}\) \(\\{\mathrm{B}\\}(4 / 3) \mathrm{MR}^{2}\) \(\\{\mathrm{C}\\}(32 / 3) \mathrm{MR}^{2}\) \(\\{\mathrm{D}\\}(34 / 3) \mathrm{MR}^{2}\)

A rod of length L rotate about an axis passing through its centre and normal to its length with an angular velocity \(\omega\). If A is the cross-section and \(D\) is the density of material of rod. Find its rotational \(\mathrm{K} . \mathrm{E}\). \(\\{\mathrm{A}\\}(1 / 2) \mathrm{AL}^{3} \mathrm{D} \omega^{2}\) \\{B \(\\}(1 / 6) \mathrm{AL}^{3} \mathrm{D} \omega^{2}\) \(\\{C\\}(1 / 24) A L^{3} D \omega^{2}\) \(\\{\mathrm{D}\\}(1 / 12) \mathrm{AL}^{3} \mathrm{D} \omega^{2}\)

Initial angular velocity of a circular disc of mass \(\mathrm{M}\) is \(\omega_{1}\) Then two spheres of mass \(\mathrm{m}\) are attached gently two diametrically opposite points on the edge of the disc what is the final angular velocity of the disc? \(\\{\mathrm{A}\\}[(\mathrm{M}+\mathrm{m}) / \mathrm{M}] \omega_{1}\) \(\\{\mathrm{B}\\}[(\mathrm{M}+4 \mathrm{~m}) / \mathrm{M}] \omega_{1}\) \(\\{\mathrm{C}\\}[\mathrm{M} /(\mathrm{M}+4 \mathrm{~m})] \omega_{1}\) \\{D\\} \([\mathrm{M} /(\mathrm{M}+2 \mathrm{~m})] \omega_{1}\)

A circular disc \(\mathrm{x}\) of radius \(\mathrm{R}\) is made from an iron plate of thickness \(t\). and another disc \(Y\) of radius \(4 R\) is made from an iron plate of thickness \(t / 4\) then the rotation between the moment of inertia \(\mathrm{I}_{\mathrm{x}}\) and \(\mathrm{I}_{\mathrm{y}}\) is \(\\{\mathrm{A}\\} \mathrm{I}_{\mathrm{y}}=64 \mathrm{I}_{\mathrm{x}}\) \(\\{B\\} I_{y}=32 I_{x}\) \(\\{\mathrm{C}\\} \mathrm{I}_{\mathrm{y}}=16 \mathrm{I}_{\mathrm{x}}\) \(\\{\mathrm{D}\\} \mathrm{I}_{\mathrm{y}}=\mathrm{I}_{\mathrm{x}}\)

A Pulley of radius \(2 \mathrm{~m}\) is rotated about its axis by a force \(F=\left(20 t-5 t^{2}\right) N\) where \(t\) is in sec applied tangentially. If the moment of inertia of the Pulley about its axis of rotation is \(10 \mathrm{KgM}^{2}\), the number of rotations made by the pulley before its direction of motion is reversed is : \(\\{\mathrm{A}\\}\) more than 3 but less then 6 \(\\{\mathrm{B}\\}\) more than 6 but less then 9 \(\\{\mathrm{C}\\}\) more than 9 \\{D \\} Less then 3

Two spheres each of mass \(\mathrm{M}\) and radius \(\mathrm{R} / 2\) are connected with a mass less rod of length \(2 \mathrm{R}\) as shown in figure. What will be moment of inertia of the system about an axis passing through centre of one of the spheres and perpendicular to the rod? \(\\{\mathrm{A}\\}(21 / 5) \mathrm{MR}^{2}\) \(\\{\mathrm{B}\\}(2 / 5) \mathrm{MR}^{2}\) \(\\{\mathrm{C}\\}(5 / 2) \mathrm{MR}^{2}\) \(\\{\mathrm{D}\\}(5 / 21) \mathrm{MR}^{2}\)

A cord is wound round the circumference of wheel of radius r. the axis of the wheel is horizontal and moment of inertia about it is I A weight \(\mathrm{mg}\) is attached to the end of the cord and falls from the rest. After falling through the distance \(\mathrm{h}\). the angular velocity of the wheel will be.... \(\\{B\\}\left[2 m g h /\left(I+m r^{2}\right)\right]\) \(\\{\mathrm{C}\\}\left[2 \mathrm{mgh} /\left(\mathrm{I}+\mathrm{mr}^{2}\right)\right]^{1 / 2}\) \(\\{\mathrm{D}\\} \sqrt{(2 \mathrm{gh})}\)

A meter stick of mass \(400 \mathrm{gm}\) is pivoted at one end and displaced through an angle 600 the increase in its P.E. is \(\overline{\\{\mathrm{A}\\} 2}\) \(\\{B\\} 3\) \(\\{\) C \(\\}\) Zero \(\\{\mathrm{D}\\} 1\)

A gramophone record of mass \(\mathrm{M}\) and radius \(\mathrm{R}\) is rotating with angular speed \(\mathrm{W}\). If two pieces of wax each of mass \(\mathrm{M}\) are kept on it at a distance of \(R / 2\) from the centre on opposite side then the new angular velocity will be..... \(\\{\mathrm{A}\\}(\omega / 2)\) \(\\{B\\}[m \omega /(M+m)\) \(\\{C\\}[M \omega /(M+m)]\) \(\\{\mathrm{D}\\}[(\mathrm{M}+\mathrm{m}) / \mathrm{M} \omega]\)

The M.I of a disc of mass \(\mathrm{M}\) and radius \(\mathrm{R}\) about an axis passing through the centre \(\mathrm{O}\) and perpendicular to the plane of disc is \(\left(\mathrm{MR}^{2} / 2\right)\). If one quarter of the disc is removed the new moment of inertia of disc will be..... \(\\{\mathrm{A}\\}\left(\mathrm{MR}^{2} / 3\right)\) \(\\{B\\}\left(M R^{2} / 4\right)\) \(\\{\mathrm{C}\\}(3 / 8) \mathrm{MR}^{2}\) \(\\{\mathrm{D}\\}(3 / 2) \mathrm{MR}^{2}\)