Dynamics of Circular Motion

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JEE Main 2020

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

\frac{m l ω ^{2}}{k + mω}

(B)

\frac{m l ω ^{2}}{k + m ω ^{2}}

(C)

\frac{m l ω ^{2}}{k - m ω ^{2}}

(D)

\frac{m l ω ^{2}}{k - mω}

JEE Main 2020

LEVELJEE Main

A spring mass system (mass $m$ , spring constant $k$ and natural length $l$ ) rests in equilibrium on 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 it's 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

(A)

\frac{m ω ^{2}}{3 k}

(B)

\frac{2 m ω ^{2}}{k}

(C)

\frac{2}{3} (\frac{m ω ^{2}}{k})

(D)

\frac{m ω ^{2}}{k}

JEE Main 2020

LEVELJEE Advanced

A bead of mass $m$ stays at point $P (a, b)$ on a wire bent in the shape of a parabola $y = 4 C x^{2}$ and rotating with angular speed $ω$ (see figure). The value of $ω$ is (neglect friction)

(A)

2 2 g C

(B)

\frac{2 g}{C}

(C)

2 g C

(D)

\frac{2 g C}{ab}

JEE Main 2021, 20 July Shift-I

LEVELJEE Advanced

The normal reaction $N$ for a vehicle of $800 kg$ mass, negotiating a turn on a $3 0^{\circ}$ banked road at maximum possible speed without skidding is $\dots \times 1 0^{3} kg-m/s^{2}$ . [Take, $μ_{s} = 0.2$ ]

JEE Main 2021, 20 July Shift-II

LEVELJEE Main

Consider a binary star system of star A and star B with masses $m_{A}$ and $m_{B}$ revolving in a circular orbit of radii $r_{A}$ and $r_{B}$ , respectively. If $T_{A}$ and $T_{B}$ are the time period of star A and star B respectively, then

(A)

T_{A} = T_{B} (\frac{r _{A}}{r _{B}})^{\frac{3}{2}}

(B)

T_{A} = T_{B}

(C)

T_{A} > T_{B}

(if

m_{A} > m_{B}

)

(D)

T_{A} > T_{B}

(if

r_{A} > r_{B}

)

JEE Main 2021, 26 Aug Shift-II

LEVELJEE Main

A particle of mass $m$ is suspended from a ceiling through a string of length $L$ . The particle moves in a horizontal circle of radius $r$ such that $r = \frac{L}{2}$ . The speed of particle will be

(A)

r g

(B)

2 r g

(C)

2 r g

(D)

\frac{r g}{2}

JEE Main 2021, 16 March Shift-II

LEVELJEE Advanced

Statement I A cyclist is moving on an unbanked road with a speed of $7 kmh^{- 1}$ and takes a sharp circular turn along a path of radius of 2 m without reducing the spee(d) The static friction coefficient is 0.2. The cyclist will not slip and pass the curve ( $g = 9.8 m / s^{2}$ ) Statement II If the road is banked at an angle of $4 5^{\circ}$ , cyclist can cross the curve of 2 m radius with the speed of $18.5 kmh^{- 1}$ without slipping. In the light of the above statements, choose the correct answer from the options given below.

(A)

Statement I is false and statement II is true.

(B)

Statement I is true and statement II is false.

(C)

Both statement I and statement II are false.

(D)

Both statement I and statement II are true.

JEE Main 2021 (17 March Shift-I)

LEVELJEE Advanced

A modern grand-prix racing car of mass $m$ is travelling on a flat track in a circular arc of radius $R$ with a speed $v$ . If the coefficient of static friction between the tyres and the track is $μ_{s}$ , then the magnitude of negative lift $f_{L}$ acting downwards on the car is (Assume forces on the four tyres are identical and $g = acceleration due to gravity$ )

(A)

m (\frac{v ^{2}}{μ _{s} R} + g)

(B)

m (\frac{v ^{2}}{μ _{s} R} - g)

(C)

m (g - \frac{v ^{2}}{μ _{s} R})

(D)

- m (g + \frac{v ^{2}}{μ _{s} R})

JEE Main 2019, 12 April Shift-II

LEVELJEE Advanced

A smooth wire of length $2 π r$ is bent into a circle and kept in a vertical plane. A bead can slide smoothly on the wire. When the circle is rotating with angular speed $ω$ about the vertical diameter $A B$ , as shown in figure, the bead is at rest with respect to the circular ring at position $P$ as shown. Then, the value of $ω^{2}$ is equal to

(A)

\frac{3 g}{2 r}

(B)

2 g / (r 3)

(C)

(g 3) / r

(D)

2 g / r

LEVELJEE Main

The minimum velocity (in $ms^{- 1}$ ) with which a car driver must traverse a flat curve of radius $150 m$ and coefficient of friction $0.6$ to avoid skidding is

A particle is moving with a uniform speed in a circular orbit of radius $R$ in a central force inversely proportional to the nth power of $R$ . If the period of rotation of the particle is $T$ , then :

(A)

T \propto R^{3/2}

for any

n

(B)

T \propto R^{\frac{n}{2} + 1}

(C)

T \propto R^{(n + 1) /2}

(D)

T \propto R^{n /2}

JEE Main 2021, 16 March Shift-I

LEVELJEE Main

A block of $200 g$ mass moves with a uniform speed in a horizontal circular groove, with vertical side walls of radius $20 cm$ . If the block takes $40 s$ to complete one round, the normal force by the side walls of the groove is

(A)

0.0314 N

(B)

9.859 \times 1 0^{- 2} N

(C)

6.28 \times 1 0^{- 3} N

(D)

9.859 \times 1 0^{- 4} N

JEE Main 2021 (26 Feb Shift-I)

LEVELJEE Main

A particle is moving with uniform speed along the circumference of a circle of radius $R$ under the action of a central fictitious force $F$ which is inversely proportional to $R^{3}$ . Its time period of revolution will be given by

(A)

T \propto R^{2}

(B)

T \propto R^{3/2}

(C)

T \propto R^{5/2}

(D)

T \propto R^{4/3}

JEE Advanced (1987)

LEVELJEE Main

A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane. It follows that

* Multiple Correct Options

(A)

its velocity is constant

(B)

its acceleration is constant

(C)

its kinetic energy is constant

(D)

it moves in a circular path

LEVELJEE Main

A car is moving in a circular horizontal track of radius $10 m$ with a constant speed of $10 m/s$ . A plumb bob is suspended from the roof of the car by a light rigid rod. The angle made by the rod with the vertical is (Take $g = 10 m/s^{2}$ )

(A)

zero

(B)

3 0^{\circ}

(C)

4 5^{\circ}

(D)

6 0^{\circ}

LEVELJEE Advanced

A long horizontal rod has a bead which can slide along its length and is initially placed at a distance $L$ from one end $A$ of the rod. The rod is set in angular motion about $A$ with a constant angular acceleration $α$ . If the coefficient of friction between the rod and bead is $μ$ , and gravity is neglected, then the time after which the bead starts slipping is

(A)

\frac{μ}{α}

(B)

\frac{μ}{α}

(C)

\frac{1}{μα}

(D)

infinitesimal

LEVELJEE Advanced

A smooth semicircular wire track of radius $R$ is fixed in a vertical plane (figure). One end of a massless spring of natural length $\frac{3 R}{4}$ is attached to the lowest point $O$ of the wire track. A small ring of mass $m$ which can slide on the track is attached to the other end of the spring. The ring is held stationary at point $P$ such that the spring makes an angle $6 0^{\circ}$ with the vertical. The spring constant $k = \frac{m g}{R}$ . Consider the instant when the ring is making an angle $6 0^{\circ}$ with the vertical. The spring is released, (a) Draw the free body diagram of the ring. (b) Determine the tangential acceleration of the ring and the normal reaction. (1996)

LEVELJEE Advanced

A small block is shot into each of the four smooth tracks as shown below. Each of the tracks rises to the same height. The speed with which the block enters the track is the same in all cases. At the highest point of the track, the normal reaction is maximum in (2001)

A hemispherical bowl of radius $R = 0.1 m$ is rotating about its own axis (which is vertical) with an angular velocity $ω$ . A particle of mass $1 0^{- 2} kg$ on the frictionless inner surface of the bowl is also rotating with the same $ω$ . The particle is at a height $h$ from the bottom of the bowl. (1993) (a) Obtain the relation between $h$ and $ω$ . What is the minimum value of $ω$ needed, in order to have a non-zero value of $h$ ? (b) It is desired to measure $g$ (acceleration due to gravity) using the set-up by measuring $h$ accurately. Assuming that $R$ and $ω$ are known precisely and that the least count in the measurement of $h$ is $1 0^{- 4} m$ , what is minimum possible error $Δ g$ in the measured value of $g$ ?

LEVELJEE Main

A ball of mass ( $m$ ) $0.5 kg$ is attached to the end of a string having length ( $l$ ) $0.5 m$ . The ball is rotated on a horizontal circular path like a physical pendulum about vertical axis. The maximum tension that the string can bear is $324 N$ . The maximum possible value of angular velocity of ball (in rad/s) is

(A)

9

(B)

18

(C)

27

(D)

36

LEVELJEE Main

A simple pendulum of length $L$ and mass (bob) $M$ is oscillating in a plane about a vertical line between angular limits $- ϕ$ and $+ ϕ$ . For an angular displacement $θ$ ( $∣ θ ∣ < ϕ$ ), the tension in the string and the velocity of the bob are $T$ and $v$ respectively. The following relations hold good under the above conditions.

* Multiple Correct Options

(A)

T cos θ = M g

(B)

T - M g cos θ = \frac{M v ^{2}}{L}

(C)

The magnitude of the tangential acceleration of the bob

∣ a_{T} ∣ = g sin θ

(D)

T = M g cos θ

LEVELJEE Advanced

Two blocks of masses $m_{1} = 10 kg$ and $m_{2} = 5 kg$ connected to each other by a massless inextensible string of length $0.3 m$ are placed along a diameter of turn table. The coefficient of friction between the table and $m_{1}$ is $0.5$ while there is no friction between $m_{2}$ and the table. The table is rotating with an angular velocity of $10 rad/s$ about a vertical axis passing through its centre $O$ . The masses are placed along the diameter of table on either side of the centre $O$ such that the mass $m_{1}$ is at a distance of $0.124 m$ from $O$ . The masses are observed to be at rest with respect to an observer on the turn table.\n(a) Calculate the frictional force on $m_{1}$ .\n(b) What should be the minimum angular speed of the turn table, so that the masses will slip from this position?\n(c) How should the masses be placed with the string remaining taut so that there is no frictional force acting on the mass $m_{1}$ ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A bead of mass m stays at point P(a,b) on a wire bent in the shape of a parabola y=4Cx2 and rotating with angular speed ω (see figure). The value of ω is (neglect friction)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The minimum velocity (in ms−1) with which a car driver must traverse a flat curve of radius 150 m and coefficient of friction 0.6 to avoid skidding is

A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane. It follows that

A small block is shot into each of the four smooth tracks as shown below. Each of the tracks rises to the same height. The speed with which the block enters the track is the same in all cases. At the highest point of the track, the normal reaction is maximum in (2001)

A bead of mass $m$ stays at point $P (a, b)$ on a wire bent in the shape of a parabola $y = 4 C x^{2}$ and rotating with angular speed $ω$ (see figure). The value of $ω$ is (neglect friction)

The minimum velocity (in $ms^{- 1}$ ) with which a car driver must traverse a flat curve of radius $150 m$ and coefficient of friction $0.6$ to avoid skidding is