Centre of Gravity

Q. centre of mass of thre particles of masses 1 kg , 2kg and 3 kg lies at pt(1, 2, 3) and CM of another system of particle of total mass3 kg lies at (-1, 3, -2). where should we put a particles of mass 5 kg so that the CM of entire system lies at the the CM of the first system

Q. 20.The reduced mass of two particles having masses m and 2m is ?

Q. The co-ordinates of a triangle PQR are P(1, 2), Q(4, 6) and R(–2, –2). Three particles of masses 2 kg, 4 kg and m kg are placed at the vertices of the triangle. If the co-ordinates of the centre of mass are calculate the mass m.

Q. 17. Consider a system of two particles having masses2 kg and 5 kg. If the particle of mass 2 kg ispushed towards the centre of mass of particlesthrough a distance 5 m, by what distance wouldparticle of mass 5 kg move so as to keep thecentre of mass of particles at the original position?(1) 2 m(2) 12.5 m(3) 8 m(4) 5 mlinear speed of a narticle moving in a circular path

Q. 2. Three particles A, B and C each of mass m arethe comers of an equilateral triangle of sideL. If the particle A is released keeping the particlesand C fixed, the magnitude of instantaneousacceleration of A is

Q. three particles of masses m , m , and 4 kg are kept on the verticals of triangle ABC . coordinate of A , B, C are (1, 2) , (3, 2) , (-2, -2) respectively such that the centre of mass lies at origin . find value of mass m

Q. 19. Expression for gravitational potential

Q. Two masses m and m/2 are connected at the two ends of a massless rigid rod of length l. The rod is suspended by a thin wire of torsional constant k at the centre of mass of rod-mass system . Because of torsional constant k, the restoring torque is equal to kθ for angular displacement θ. If the rod is rotated by θ0 and released the tension in it, when it passes through its mean position will be

Q. three particles each of mass m are placed at the vertices of an equilateral triangle of side a . the gravitational field intensity at the centroid of the triangle u

Q. A simple pendulum of length $100\text{cm}$ is suspended from the ceiling of a cart which is sliding without friction on an inclined plane of inclination ${60}^{\circ }$. Find the time period of the pendulum.
Take $g=10{\text{m/s}}^2$.

1. $2\text{s}$
2. $2.8\text{s}$
3. $5\text{s}$
4. $2.14\text{s}$

Q.

A carpet of length 10 m is spread on horizontal ground. Now if it is folded to double on it self, the shift in the center of mass of the carpet with respect to the original center of mass is

1. 2.5cm

2. 5 m

3. 7.5m

4. 1.25m

Q. 3 equal.weights of mass 2 kg each are hnging on a string passing over a fixed frictionless pulley The tendion in string AB

Q. A man measures the time period of a simple pendulum inside a stationary lift and finds it to be $10\text{sec}$. If the lift accelerates upwards with an acceleration $\frac{g}{4}$, then find the time period of the pendulum.

1. $8\sqrt{5}\text{sec}$
2. $4\sqrt{5}\text{sec}$
3. $10\sqrt{5}\text{sec}$
4. $2\sqrt{5}\text{sec}$

Q. Seven homogeneous bricks, each of length L, are arranged as shown in figure. Each brick is displaced with respect to the one in contact by L/10. Find the x-coordinate fo the centre of mass relative to the origin shown.
Figure

Q. The period of oscillation of a simple pendulum of length L suspended from the roof of a vehicle which moves without friction down in an inclined plane of inclination $\alpha ,$ is given by

1. $2\pi \sqrt{\frac{L}{gcos\alpha }}$
2. $2\pi \sqrt{\frac{L}{gsin\alpha }}$
3. $2\pi \sqrt{\frac{L}{g}}$
4. $2\pi \sqrt{\frac{L}{gtan\alpha }}$

Q. The length of a simple pendulum is $32\text{cm}$. It is suspended by the roof of a lift which is moving upwards with an acceleration of $6.2{\text{m/s}}^2$. Find the time period of pendulum.

1. $0.2\sqrt{2}\pi \text{s}$
2. $\sqrt{2}\pi \text{s}$
3. $0.6\sqrt{2}\pi \text{s}$
4. $0.2\sqrt{6}\pi \text{s}$

Q. If all the particles of a system lie in X-Y plane, is it necessary that the centre of mass be in X-Y plane?

Q.

A simple pendulum has time period $T_1$. The point of suspension is now moved upward according to the relation $y=k{t}^2,(k=\frac{1m}{s^2})$ where y is the vertical displacement. The time period now becomes $T_2$. The ratio of$\frac{T_1^2}{T_2^2}$ is (Take g =$10\frac{m}{s^2})$

1. $\frac{6}{5}$

2. $\frac{5}{6}$

3. 1

4. $\frac{4}{5}$

Q.

The time period of a simple oscillation is measured as T in a stationary lift. If the lift moves upwards with an acceleration of 5 g, the time period will be

1. same

2. increased by $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.$ times

3. decreased by $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ times

4. none of the above.

Q. A simple pendulum of length $l$ and bob mass m is displaced from its equilibrium position O to a position P so that the height of P above O is $h$ and is released. What is the tension in the string when the bob passes through the equilibrium position O? Neglect friction. V is the velocity of the bob at O

1. $m(g+\frac{V^2}{l})$
2. $\frac{2mgh}{l}$
3. $mg(1+\frac{h}{l})$
4. $mg(1+\frac{2h}{l})$

Q. an isolated particle of mass m moving in horizontal plane [x-y] along the x axis at a certain height above the ground . it suddenly explodes into fragments of masses m/4 and 3m/4 . an instant later smaller fragment is at y=+15 cm . the larger fragment at this instant is at

Q. gravitational potential in a region as V=x^2yz j/kg fing I at 1, 2, 1

Q. 1.If a hollow sphere is released at the place of ring from it will strike the ground further from ring. 2. If disc is released it will strike at a point 2R/\sqrt{}3 from the base. 3. If mass and radius of ring are doubled then it will strike at distance R/2 from the base. 4. If a solid sphere is released from the top it will strike at a point R\sqrt{}(10/7) from the base.

Q. Two point particles of mass m and 2m are initially seperated by a dis†an ce 4a . They are then released to become free to move . Find the velocities of both the particles when dis†an ce between them reduces to a