Centre of Mass & Rotational Motion · JEE Main previous year questions

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2020 30 questions

Q262020 · 6 Sept 2020 · Shift 1
1. A. $3 m l^{2} ω$
2. B. $4 m l^{2} ω$
3. C. $m l^{2} ω$
4. D. $2 m l^{2} ω$
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Answer: A
Q302020 · 6 Sept 2020 · Shift 1
1. A. $\frac{M ( R ^{2} + H ^{2} )}{3}$
2. B. $\frac{M R ^{2}}{2}$
3. C. $\frac{M R ^{2}}{3}$
4. D. $\frac{M H ^{2}}{3}$
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Answer: B
Q492020 · 6 Sept 2020 · Shift 1· Numerical answer
Initially a gas of diatomic molecules is contained in a cylinder of volume $V_{1}$ at a pressure $P_{1}$ and temperature $250 K$ . Assuming that $25%$ of the molecules get dissociated causing a change in number of moles. The pressure of the resulting gas at temperature $2000 K$ , when contained in a volume $2 V_{1}$ is given by $P_{2}$ . The ratio $\frac{P _{2}}{P _{1}}$ is ______ .
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Answer: 5
Q322020 · 6 Sept 2020 · Shift 2
A square loop of side $2 a$ and carrying current I is kept in xz plane with its centre at origin. A long wire carrying the same current I is placed parallel to z-axis and passing through point $(0, b, 0)$ , $(b ≫ a)$ . The magnitude of torque on the loop about z-axis will be :
1. A. $\frac{2 μ _{0} I ^{2} a ^{2}}{π b}$
2. B. $\frac{2 μ _{0} I ^{2} a ^{2} b}{π ( a ^{2} + b ^{2} )}$
3. C. $\frac{μ _{0} I ^{2} a ^{2} b}{2 π ( a ^{2} + b ^{2} )}$
4. D. $\frac{μ _{0} I ^{2} a ^{2}}{2 π b}$
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Answer: A
Q352020 · 6 Sept 2020 · Shift 2
The linear mass density of a thin rod AB of length L varies from A to B as $λ (x) = λ_{0} (1 + \frac{x}{L})$ , where x is the distance from A. If M is the mass of the rod then its moment of inertia about an axis passing through A and perpendicular to the rod is :
1. A. $\frac{2}{5} M L^{2}$
2. B. $\frac{5}{12} M L^{2}$
3. C. $\frac{7}{18} M L^{2}$
4. D. $\frac{3}{7} M L^{2}$
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Answer: C
Q472020 · 6 Sept 2020 · Shift 2· Numerical answer
The centre of mass of solid hemisphere of radius 8 cm is x from the centre of the flat surface. Then value of x is __________.
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Answer: 3
Q282020 · 3 Sept 2020 · Shift 1
Moment of inertia of a cylinder of mass M, length L and radius R about an axis passing through its centre and perpendicular to the axis of the cylinder is $I = M (\frac{R ^{2}}{4} + \frac{L ^{2}}{12})$ . If such a cylinder is to be made for a given mass of a material, the ratio $\frac{L}{R}$ for it to have minimum possible I is
1. A. $\frac{3}{2}$
2. B. $\frac{3}{2}$
3. C. $\frac{2}{3}$
4. D. $\frac{2}{3}$
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Answer: B
Q492020 · 3 Sept 2020 · Shift 1· Numerical answer
A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating about its axis at 5 revolutions per minute (rpm). The person now starts moving towards the centre of the platform. What will be the rotational speed (in rpm) of the platform when the person reaches its centre _________.
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Answer: 9
Q332020 · 3 Sept 2020 · Shift 2
1. A. $cos θ = \frac{2 g}{3 l ω ^{2}}$
2. B. $cos θ = \frac{3 g}{2 l ω ^{2}}$
3. C. $cos θ = \frac{g}{2 l ω ^{2}}$
4. D. $cos θ = \frac{g}{l ω ^{2}}$
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Answer: B
Q462020 · 3 Sept 2020 · Shift 2· Numerical answer
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Answer: 25
Q472020 · 3 Sept 2020 · Shift 2· Numerical answer
A galvanometer coil has 500 turns and each turn has an average area of $3 \times 1 0^{- 4} m^{2}$ . If a torque of 1.5 Nm is required to keep this coil parallel to a magnetic field when a current of 0.5 A is flowing through it, the strength of the field (in T) is _______.
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Answer: 20
Q292020 · 2 Sept 2020 · Shift 1
1. A. $\frac{2 ω ^{2}}{25 g}$
2. B. $\frac{5 ω ^{2}}{2 g}$
3. C. $\frac{25 ω ^{2}}{2 g}$
4. D. $\frac{2 ω ^{2}}{5 g}$
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Answer: C
Q332020 · 2 Sept 2020 · Shift 1
1. A. $M g 1 - (\frac{R - a}{R})^{2}$
2. B. $M g 1 - \frac{a ^{2}}{R ^{2}}$
3. C. $M g \frac{a}{R}$
4. D. $M g (\frac{R}{R - a})^{2} - 1$
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Answer: A
Q392020 · 2 Sept 2020 · Shift 2
The height 'h' at which the weight of a body will be the same as that at the same depth 'h' from the surface of the earth is (Radius of the earth is R and effect of the rotation of the earth is neglected)
1. A. $\frac{R}{2}$
2. B. $\frac{5 R - R}{2}$
3. C. $\frac{3 R - R}{2}$
4. D. $\frac{5}{2} R - R$
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Answer: B
Q402020 · 2 Sept 2020 · Shift 2
Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are $0.1 kg \cdot m^{2}$ and $10 rad s^{- 1}$ respectively while those for the second one are $0.2 kg \cdot m^{2}$ and $5 rad s^{- 1}$ respectively. At some instant they get stuck together and start rotating as a single system about their common axis with some angular speed. The kinetic energy of the combined system is :
1. A. $\frac{20}{3} J$
2. B. $\frac{5}{3} J$
3. C. $\frac{10}{3} J$
4. D. $\frac{2}{3} J$
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Answer: A
Q462020 · 2 Sept 2020 · Shift 2· Numerical answer
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Answer: 23
Q442020 · 9 Jan 2020 · Shift 1
1. A. $\frac{13}{23}$
2. B. $\frac{23}{13}$
3. C. $\frac{15}{13}$
4. D. $\frac{13}{15}$
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Answer: A
Q272020 · 9 Jan 2020 · Shift 2
A spring mass system (mass m, spring constant k and natural length $l$ ) rests in equilibrium on a horizontal disc. The free end of the spring is fixed at the centre of the disc. If the disc together with spring mass system, rotates about its axis with an angular velocity $ω$ , ( $k ≫ m ω^{2}$ ) the relative change in the length of the spring is best given by the option :
1. A. $\frac{m ω ^{2}}{3 k}$
2. B. $\frac{m ω ^{2}}{k}$
3. C. $\frac{2 m ω ^{2}}{k}$
4. D. $\frac{2}{3} (\frac{m ω ^{2}}{k})$
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Answer: B
Q282020 · 9 Jan 2020 · Shift 2
A rod of length L has non-uniform linear mass density given by $ρ (x) = a + b (\frac{x}{L})^{2}$ , where a and b are constants and $0 \leq x \leq L$ . The value of x for the centre of mass of the rod is at :
1. A. $\frac{3}{2} (\frac{a + b}{2 a + b}) L$
2. B. $\frac{4}{3} (\frac{a + b}{2 a + 3 b}) L$
3. C. $\frac{3}{4} (\frac{2 a + b}{3 a + b}) L$
4. D. $\frac{3}{2} (\frac{2 a + b}{3 a + b}) L$
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Answer: C
Q412020 · 9 Jan 2020 · Shift 2
1. A. $[\frac{2 ( m _{1} + m _{2} ) g h}{( m _{1} + m _{2} ) R ^{2} + I}]^{\frac{1}{2}}$
2. B. $[\frac{m _{1} + m _{2}}{( m _{1} + m _{2} ) R ^{2} + I}]^{\frac{1}{2}} g h$
3. C. $[\frac{( m _{1} - m _{2} )}{( m _{1} + m _{2} ) R ^{2} + I}]^{\frac{1}{2}} g h$
4. D. $[\frac{2 ( m _{1} - m _{2} ) g h}{( m _{1} + m _{2} ) R ^{2} + I}]^{\frac{1}{2}}$
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Answer: D
Q302020 · 8 Jan 2020 · Shift 1
Consider a uniform rod of mass $M = 4 m$ and length $ℓ$ pivoted about its centre. A mass m moving with velocity v making angle $θ = \frac{π}{4}$ to the rod's long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is :
1. A. $\frac{3 2}{7} \frac{v}{ℓ}$
2. B. $\frac{3}{7} \frac{v}{ℓ}$
3. C. $\frac{3}{7 2} \frac{v}{ℓ}$
4. D. $\frac{4}{7} \frac{v}{ℓ}$
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Answer: A
Q352020 · 8 Jan 2020 · Shift 1
1. A.(0.75m, 1.75m)
2. B.(1m, 1.75m)
3. C.(0.75m, 0.75m)
4. D.(1.25m, 1.50m)
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Answer: A
Q382020 · 8 Jan 2020 · Shift 1
A particle of mass m is fixed to one end of a light spring having force constant k and unstretched length $ℓ$ . The other end is fixed. The system is given an angular speed $ω$ about the fixed end of the spring such that it rotates in a circle in gravity free space. Then the stretch in the spring is :
1. A. $\frac{m ℓ ω ^{2}}{k - m ω ^{2}}$
2. B. $\frac{m ℓ ω ^{2}}{k - mω}$
3. C. $\frac{m ℓ ω ^{2}}{k + m ω ^{2}}$
4. D. $\frac{m ℓ ω ^{2}}{k + mω}$
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Answer: A
Q402020 · 8 Jan 2020 · Shift 1
A leak proof cylinder of length 1m, made of a metal which has very low coefficient of expansion is floating vertically in water at 0°C such that its height above the water surface is 20 cm. When the temperature of water is increased to 4°C, the height of the cylinder above the water surface becomes 21 cm. The density of water at T = 4°C, relative to the density at T = 0°C is close to :
1. A.1.04
2. B.1.26
3. C.1.01
4. D.1.03
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Answer: C
Q492020 · 8 Jan 2020 · Shift 1· Numerical answer
A particle is moving along the x-axis with its coordinate with the time 't' given be $x (t) = 10 + 8 t - 3 t^{2}$ . Another particle is moving the y-axis with its coordinate as a function of time given by $y (t) = 5 - 8 t^{3}$ . At t = 1s, the speed of the second particle as measured in the frame of the first particle is given as $v$ . Then v (in m/s) is _______.
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Answer: 580