Q21JEE Main 2023NAT

The moment of inertia of a semicircular ring about an axis, passing through the center and perpendicular to the plane of ring, is $\;\frac{1}{x} M R^2$ , where $R$ is the radius and $M$ is the mass of the semicircular ring. The value of $x$ will be _______. [8-Apr-2023 shift 1]

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📖 Explanation

$\;I=∫ d m R^2 \Rightarrow R^2 ∫ {\text{dm}}= {\text{MR}}^2$ $\; {\text{I}}= {\text{MR}}^2$ $\;\;\text{ Given }\; {\text{I}}=\;{1}/{ {\text{x}}} M {\text{M}}^2$ $\; {\text{x}}=1$

Q22JEE Main 2023MCQ

Given below are two statements : one is labelled as Assertion $\text{bo} {A}$ and then other is labelled as Reason $\text{bo} {R}$ Assertion A : An electric fan continues to rotate for some time after the current is switched off.Reason $\text{bo} {R}$ : Fan continues to rotate due to inertia of motion.In the light of above statements, choose the most appropriate answer from the options given below. [10-Apr-2023 shift 2]

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Q23JEE Main 2023NAT

A solid cylinder is released from rest from the top of an inclined plane of inclination $30^{\circ}$ and length $60 {\text{cm}}$ . If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is ________ ${\text{ms}}^{-1}$ . (Given ${\text{g}}=10 {\text{ms}}^{-2}$ ) [1-Feb-2023 Shift 1]

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📖 Explanation

$v=\sqrt{\;\frac{2 g h}{1+\;\frac{k^2}{R^2}}}$ Where $h=60 sin \;30^{\circ}=30 {\text{cm}}$ $k^2=\;\frac{R^2}{2}$ $v=2 {\text{ms}}^{-1}$

Q24JEE Main 2023NAT

Two discs of same mass and different radii are made of different materials such that their thicknesses are $1 {\text{cm}}$ and $0.5 {\text{cm}}$ respectively. The densities of materials are in the ratio 3:5. The moment of inertia of these discs respectively about their diameters will be in the ratio of $-$ . The value of $x$ is _______. [31-Jan-2023 Shift 2]

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📖 Explanation

$m=\rho \pi R^2 t$ $\;\text{ so }\; R^2=\;\frac{m}{\rho \pi t}$ $I=\;\frac{m R^2}{4}=\;\frac{m^2}{4 \rho \pi t}$ So $\;\frac{I_1}{I_2}=\;\frac{\rho _2 t_2}{\rho _1 t_1}=\;\frac{5}{3} \times \;\frac{0.5}{1}=\;\frac{5}{6}$ So $x=5$

Q25JEE Main 2023NAT

A ring and a solid sphere rotating about an axis passing trough their centers have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is $\sqrt{\;\frac{2}{x}}$ . The value of $x$ is _______. [6-Apr-2023 shift 2]

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📖 Explanation

Given radius of gyration is same for ring and solid sphere $\; {\text{K}}_{ {\text{R}}}= {\text{K}}_{ {\ ss}}$ $\; {\text{R}}_{ {\text{R}}}=\sqrt{\;\frac{2}{5}} {\text{R}}_{ {\ ss}}$ $\;\;\text{ or }\; \;{ {\text{R}}_{ {\text{R}}}}/{ {\text{R}}_{ {\ ss}}}=\sqrt{\;\frac{2}{5}}$ $\;\;\text{ therefore }\; {\text{x}}=5$

Q26JEE Main 2023NAT

A hollow spherical ball of uniform density rolls up a curved surface with an initial velocity $3 {\text{m}} / {\ s}$ (as shown in figure). Maximum height with respect to the initial position covered by it will be _______ ${\text{cm}} (.$ take, $g=10 {\text{m}} / {\ s}^2)$ [8-Apr-2023 shift 2]

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📖 Explanation

For pure rolling $( {\text{M}} . {\text{E}})_{ {\text{A}}}=( {\text{M}} . {\text{E}})_{ {\text{B}}}$ $\;\Rightarrow \;\frac{1}{2} {\ mv}_0^{2+}\;\frac{1}{2} \times (\;\frac{2}{3} {\text{mR}}^2) (\;{ {\text{v}}_0}/{ {\text{R}}}) ^2= {\text{mgH}}_{\max }$ $\;\Rightarrow {\text{H}}_{\max }=\;\frac{5}{6} \;{ {\text{v}}_0^2}/{ {\text{g}}}=\;\frac{5}{6} \times \;\frac{3^2}{10}=0.75 {\text{m}}$ $\;\Rightarrow {\text{H}}_{\max }=75 {\text{cm}}$

Q27JEE Main 2023NAT

A solid sphere of mass $2 {\text{kg}}$ is making pure rolling on a horizontal surface with kinetic energy $2240 {\text{J}}$ . The velocity of centre of mass of the sphere will be _______ ${\text{ms}}^{-1}$ . [29-Jan-2023 Shift 1]

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📖 Explanation

$\; {\text{KE}}=\;\frac{1}{2} {\ mv}^{2+}\;\frac{1}{2} {\text{I}} \omega ^2$ $\;2240=\;\frac{1}{2} 2( {\text{v}})^{2+}\;\frac{1}{2} \;\frac{2}{5}(2) {\text{R}}^2 \cdot (\;{ {\text{v}}}/{ {\text{R}}}) ^2$ $\;2240= {\text{v}}^{2+}\;\frac{2}{5} {\text{v}}^2$ $\;\Rightarrow {\text{v}}=40 {\text{m}} / {\ s}$

Q28JEE Main 2023MCQ

An object of mass $8 {\text{kg}}$ is hanging from one end of a uniform rod ${\text{CD}}$ of mass $2 {\text{kg}}$ and length $1 {\text{m}}$ pivoted at its end ${\text{C}}$ on a vertical wall as shown in figure. It is supported by a cable ${\text{AB}}$ such that the system is in equilibrium. The tension in the cable is :(Take ${\text{g}}=10 {\text{m}} / {\ s}^2$ ) [25-Jan-2023 Shift 1]

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📖 Explanation

Taking torque about point ${\text{C}}$ $\;\;{ {\text{T}}}/{2} \times 60=20 \times 50+80 \times 100$ $\;\Rightarrow 3 {\text{T}}=100+800$ $\;\Rightarrow =300 {\text{N}}$

Q29JEE Main 2023NAT

Two identical solid spheres each of mass $2 {\text{kg}}$ and radii $10 {\text{cm}}$ are fixed at the ends of a light rod. The separation between the centres of the spheres is $40 {\text{cm}}$ . The moment of inertia of the system about an axis perpendicular to the rod passing through its middle point is _______ $\times 10^{-3} {\text{kg}}- {\text{m}}^2$ [6-Apr-2023 shift 1]

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📖 Explanation

Using parallel axix theorem, $\;I_{\;\text{sys }\;}= (\;\frac{2}{5} m r^{2+}m d^2) \times 2$ $\;\Rightarrow I_{\;\text{sys }\;}= (\;\frac{2}{5} \times 2 \times 0.01+2 \times 0.04) \times 2=(0.008+0.08) \times 2=0.088 \times 2=176 \times 10^{-3}$

Q30JEE Main 2023NAT

A uniform solid cylinder with radius $R$ and length ${\text{L}}$ has moment of inertia ${\text{I}}_1$ , about the axis of cylinder. A concentric solid cylinder of radius ${\text{R}}^{'}=\;{ {\text{R}}}/{2}$ and length ${\text{L}}^{'}=\;{ {\text{L}}}/{2}$ is caned out of the original cylinder. If ${\text{I}}_2$ is the moment of inertia of the carved out portion ot the cylinder then $\;{ {\text{I}}_1}/{ {\text{I}}_2}=$ ___ [24-Jan-2023 Shift 2]

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📖 Explanation

$\;I_1=\;\frac{m_1 R^2}{2} \;\; I_2=\;{m_2( {\text{R}} / 2)^2}/{2}$ $\;\;\frac{I_1}{I_2}=\;\frac{4 m_1}{m_2}=\;\frac{4 \cdot \rho \pi R^2 ℓ}{\rho \cdot \;\frac{\pi R^2}{4} \times \;\frac{ℓ}{2}} \Rightarrow \;\frac{I_1}{I_2}=32$

Q31JEE Main 2023NAT

A light rope is wound around a hollow cylinder of mass $5 {\text{kg}}$ and radius $70 {\text{cm}}$ . The rope is pulled with a force of $52.5 {\text{N}}$ . The angular acceleration of the cylinder will be _______ ${\text{rads}}^{-2}$ . [13-Apr-2023 shift 2]

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📖 Explanation

$\;\tau = {\text{l}} \alpha$ $\; {\text{FR}}= {\text{mR}}^2 \alpha$ $\;\propto =\;{ {\text{F}}}/{ {\text{mR}}}=\;\frac{52.5}{5 \times 0.7}=15 {\text{rads}}^{-2}$

Q32JEE Main 2023NAT

Solid sphere $A$ is rotating about an axis PQ. If the radius of the sphere is $5 {\text{cm}}$ then its radius of gyration about PQ will be ${\sqrt{x}} {\text{cm}}$ . The value of $x$ is___ [24-Jan-2023 Shift 1]

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📖 Explanation

$\; {\text{I}}_{ {\text{cm}}}=\;\frac{2}{5} {\text{MR}}^2$ $\; {\text{I}}_{ {\text{PQ}}}= {\text{I}}_{ {\text{cm}}}+ {\text{md}}^2$ $\; {\text{I}}_{ {\text{PQ}}}=\;\frac{2}{5} {\text{mR}}^{2+} {\text{m}}(10 {\text{cm}})^2$ For radius of gyration $\; {\text{I}}_{ {\text{PQ}}}= {\text{mk}}^2$ $\; {\text{k}}^2=\;\frac{2}{5} {\text{R}}^{2+}(10 {\text{cm}})^2$ $\;=\;\frac{2}{5}(5)^{2+}100$ $\;=10+100=110$ $\; {\text{k}}={\sqrt{110}} {\text{cm}}$ $\; {\text{x}}=110$

Q33JEE Main 2023NAT

${\text{I}}_{ {\text{CM}}}$ is moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc. ${\text{I}}_{ {\text{AB}}}$ is it's moment of inertia about an axis ${\text{AB}}$ perpendicular to plane and parallel to axis ${\text{CM}}$ at a distance $\;\frac{2}{3} {\text{R}}$ from center. Where ${\text{R}}$ is the radius of the disc. The ratio of ${\text{I}}_{ {\text{AB}}}$ and ${\text{I}}_{ {\text{CM}}}$ is $x: 9$ . The value of $x$ is ________. [25-Jan-2023 Shift 1]

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📖 Explanation

$\; {\text{I}}_{ {\text{cm}}}=\;{ {\text{mR}}^2}/{2}$ $\; {\text{I}}_{ {\text{AB}}}=\;{ {\text{mR}}^2}/{2}+ {\text{m}} (\;{2 {\text{R}}}/{3}) ^2=\;\frac{17}{18} {\text{mR}}^2$ $\;\;{ {\text{I}}_{ {\text{AB}}}}/{ {\text{I}}_{ {\text{cm}}}}=\;\frac{17}{9} \Rightarrow {\text{x}}=17$

Q34JEE Main 2023NAT

A solid sphere of mass $500 {\text{g}}$ and radius $5 {\text{cm}}$ is rotated about one of its diameter with angular speed of $10 {\text{rad}}$ ${\ s}^{-1}$ . If the moment of inertia of the sphere about its tangent is ${\text{x}} \times 10^{-2}$ times its angular momentum about the diameter. Then the value of ${\text{x}}$ will be ________. [11-Apr-2023 shift 1]

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📖 Explanation

$\; {\text{I}}_1=\;\frac{2}{5} {\text{mR}}^2$ $\; {\text{I}}_2=\;\frac{2}{5} {\text{mR}}^{2+} {\text{mR}}^2=\;\frac{7}{5} {\text{mR}}^2$ Angular moment about diameter is ${\text{L}}_{ {\text{com}}}= {\text{I}}_1 {\text{w}}=\;\frac{2}{5} {\text{mR}}^2 {\text{w}}$ Now, $\;\;{ {\text{I}}_2}/{ {\text{L}}_{\;\text{com }\;}}=\;{\;\frac{7}{5} {\text{mR}}^2}/{\frac{2}{5} {\text{mR}}^2 {\text{w}}}=\;\frac{7}{2} {\text{w}}$ $\;\;{ {\text{I}}_2}/{ {\text{L}}_{\;\text{com }\;}}=\;\frac{7}{2 \times 10}=\;\frac{7}{20}$ Now $\;\frac{7}{20}= {\text{x}} \times 10^{-2}$ $\;x=\;\frac{7}{20} \times 100$ $\;x=35$ Ans.

Q35JEE Main 2023NAT

A solid sphere and a solid cylinder of same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyrations respectively $( {\text{k}}_{ {\ sph}}: {\text{k}}_{ {\text{eyl}}})$ is $2: \sqrt{ {\text{x}}}$ . The value of ${\text{x}}$ is _______. [15-Apr-2023 shift 1]

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📖 Explanation

$\; {\text{I}}_{\;\text{sphere }\;}=\;\frac{2}{5} {\text{MR}}^2= {\text{Mk}}^2$ $\; {\text{k}}_{ {\ sph}}=\sqrt{\;\frac{2}{5}} {\text{R}}$ $\; {\text{I}}_{\;\text{cylinder }\;}=\;{ {\text{MR}}^2}/{2}= {\text{Mk}}^2$ $\; {\text{k}}_{\;\text{cyl }\;}=\;{ {\text{R}}}/{{\sqrt{2}}}$ $\;\;{ {\text{k}}_{ {\ sph}}}/{ {\text{k}}_{ {\text{cyl}}}}=\;{2}/{{\sqrt{5}}}$ $\; {\text{x}}=5$

Q36JEE Main 2023NAT

A circular plate is rotating horizontal plane, about an axis passing through its center perpendicular to the plate, with an angular velocity $\omega$ . A person sits at the center having two dumbbells in his hands. When he stretches out his hands, the moment of inertia of the system becomes triple. If ${\text{E}}$ be the initial Kinetic energy of the system, then final Kinetic energy will be $\;\frac{E}{x}$ . The value of $x$ is ______ [11-Apr-2023 shift 2]

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📖 Explanation

$\; {\text{I}}_1 \omega _1= {\text{I}}_2 \omega _2$ $\; {\text{I}} \omega =3 {\text{I}} \omega _2$ $\;\omega _2=\;\frac{\omega }{3}$ $\; {\text{E}}=\;\frac{1}{2} {\text{I}} \omega ^2$ $\; {\text{E}}_{ {\text{f}}}=\;\frac{1}{2} \times 3 {\text{I}} \times (\;\frac{\omega }{3}) ^2=\;{\;\frac{1}{2} {\text{I}} \omega ^2}/{3}$ $\; {\text{x}}=3$

Q37JEE Main 2022MCQ

A solid spherical ball is rolling on a frictionless horizontal plane surface about its axis of symmetry. The ratio of rotational kinetic energy of the ball to its total kinetic energy is [26-Jun-2022-Shift-2]

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📖 Explanation

$K E_R=\;\frac{1}{2} l w^2$ $=\;\frac{1}{2} \times \;\frac{2}{5} \times \omega ^2 \times (m R^2)$ $K E_{\;\text{total }\;}=\;\frac{1}{2} \times \;\frac{7}{5} \times m R^2 \times \omega ^2$ $\therefore \;{K E_R}/{K E_{\;\text{total }\;}}=\;\frac{2}{7}$

Q38JEE Main 2022MCQ

Choose the correct answer from the options given below : List-I List-II (A) Moment of inertia of solid sphere of radius R about any tangent. (I) $\;\frac{5}{3} M R^2$ (B) Moment of inertia of hollow sphere of radius (R) about any tangent. (II) $\;\frac{7}{5} M R^2$ (C) Moment of inertia of circular ring of radius (R) about its diameter. (III) $\;\frac{1}{4} M R^2$ (D) Moment of inertia of circular disc of radius (R) about any diameter. (IV) $\;\frac{1}{2} M R^2$ [28-Jun-2022-Shift-1]

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📖 Explanation

(A) Moment of inertia of solid sphere of radius ${\text{R}}$ about a tangent $=\;\frac{2}{5} M R^{2+}M R^2=\;\frac{7}{5} M R^2$ $\Rightarrow A-(I I)$ (B) Moment of inertia of hollow sphere of radius ${\text{R}}$ about a tangent $=\;\frac{2}{3} M R^{2+}M R^2=\;\frac{5}{3} M R^2$ $\Rightarrow B-(I)$ (C) Moment of inertia of circular ring of radius $( {\text{R}})$ about its diameter $=\;\frac{ (M R^2) }{2}$ $\Rightarrow {\text{C}}-( {\text{IV}})$ (D) Moment of inertia of circular disc of radius ( $R$ ) about any diameter $=\;\frac{M R^2 / 2}{2}=\;\frac{M R^2}{4}$ $\Rightarrow {\text{D}}-( {\text{III}})$

Q39JEE Main 2022MCQ

One end of a massless spring of spring constant ${\text{k}}$ and natural length ${\text{I}}_0$ is fixed while the other end is connected to a small object of mass ${\text{m}}$ lying on a frictionless table. The spring remains horizontal on the table. If the object is made to rotate at an angular velocity $\omega$ about an axis passing through fixed end, then the elongation of the spring will be : [27-Jun-2022-Shift-2]

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📖 Explanation

$m \omega ^2 (l_0+x) =k x$ $\Rightarrow m \omega ^2 l_0= (k-m \omega ^2) \times x$ $\Rightarrow x=\;\frac{m \omega ^2 l_0}{ (k-m \omega ^2) }$

Q40JEE Main 2022NAT

A pulley of radius $1.5 {\text{m}}$ is rotated about its axis by a force $F=(12 {\text{t}}-3 {\text{t}}^2) N$ applied tangentially (while $t$ is measured in seconds). If moment of inertia of the pulley about its axis of rotation is $4.5 {\text{kg}} {\text{m}}$ , the number of rotations made by the pulley before its direction of motion is reversed, will be $\;\frac{K}{\pi }$ . The value of ${\text{K}}$ is ________. [27-Jul-2022-Shift-1]

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📖 Explanation

$\tau = {\text{I}} \alpha \Rightarrow (12 {\text{t}}-3 {\text{t}}^2) 1.5=4.5 \alpha$ $\Rightarrow \alpha =4 {\text{t}}- {\text{t}}^2$ $\Rightarrow \;{ {\text{d}} \omega }/{ {\text{dt}}}=4 {\text{t}}- {\text{t}}^2 \Rightarrow \omega =∫_{0}^{ {\text{t}}}(4 {\text{t}}- {\text{t}}^2) {\text{dt}}$ $\Rightarrow \omega =2 {\text{t}}^{2-} {\text{t}}^3 / 3$