Displacement of COM:Application

Q.

Two infinitely long parallel wires having linear charge densities ${\lambda }_1$ and ${\lambda }_2$ respectively are placed at a distance of R metres. The force per unit length on either wire will be $(K=\frac{1}{4\pi {ϵ}_0})$

1. $K\frac{2{\lambda }_1{\lambda }_2}{R}$
2. $K\frac{{\lambda }_1{\lambda }_2}{R^2}$
3. $K\frac{{\lambda }_1{\lambda }_2}{R}$
4. $K\frac{2{\lambda }_1{\lambda }_2}{R^2}$

Q.

Under the action of force, a 2 kg body moves such that x is a function of time given by $x=t^3/3$, x in metre, t in seconds, find the work done in first two seconds.

Q. A man of mass $M$ stands at one end of a plank of length $L$ which is at rest on a frictionless horizontal surface. The man walks to the other end of the plank. If mass of the plank is $\frac{M}{3}$, the distance that the man moves relative to the ground is

1. $\frac{3L}{4}$
2. $\frac{L}{3}$
3. $L$
4. $\frac{L}{4}$

Q. Two men $(A$ & $B)$ of mass $50\text{kg}$ and $100\text{kg}$ are standing at the two opposite ends of a boat of mass $200\text{kg}$ and length $L=20\text{m}$. $A$ travels a distance $5\text{m}$ right relative to the boat towards the centre and $B$ moves a distance $15\text{m}$ left relative to the boat and meets $A$. Find the distance travelled by the boat in the water when they meet.

1. $\frac{100}{7}\text{m}$
2. $\frac{75}{7}\text{m}$
3. $\frac{25}{7}\text{m}$
4. $\frac{50}{7}\text{m}$

Q.

A bomb of mass 1 kg initially at rest, explodes and breaks into three fragments of masses in the ratio 1:1:3. The two pieces of equal masses fly off perpendicular to each other with a speed 15 m/s each. What will be the speed of heavier fragment ?

Q.

Water is filled in a rectangular tank of size 3m $\times$ 2 m $\times$ 1 m.

(a) Find the total force exerted by the water on the bottom surface of the tank.

(b) Consider a vertical side of area 2 m $\times$ 1 m. Take a horizontal strip of width $\delta$x metre in this side, situated at a depth of x metre from the surface of water. Find the force by the water on this strip.

(c) Find the torque of the force calculated in part (b) about the bottom edge of this side.

(d) Find the total force by the water on this side.

(e) Find the total torque by the water on the side about the bottom edge. Neglect the atmospheric pressure and take g= $10m{s}^{-2}$.

Q.

The string, the spring and the pulley shown in figure (12-E9) are light. Find the time period of the mass m.

Q.

A shell following a parabolic path explodes somewhere in its flight. The center of mass of fragments will continue to move in

Q. Two blocks of masses $3\text{kg}$ and $2\text{kg}$ respectively are tied to the ends of a string which passes over a light pulley as shown in the figure below. The masses are held at rest at the same horizontal level and then released. The distance moved by $COM$ in $5$ seconds is

1. $10\text{m}$ upward
2. $5\text{m}$ upward
3. $5\text{m}$ downward
4. $10\text{m}$ downward

Q. A glass capillary tube is of the shape of truncated cone with an apex angle $\alpha$, so that its two ends have cross-sections of different radii. When dipped in water vertically, water rises in it to a height $h$, where the radius of its cross-section is $b$. If the surface tension of water is $S$, its density is $\rho ,$ and its contact angle with glass is $\theta .$ the value of $h$ will be ($g$ is the acceleration due to gravity).

1. $\frac{2S}{b\rho g}\cos (\theta -\alpha )$
2. $\frac{2S}{b\rho g}\cos (\theta +\alpha )$
3. $\frac{2S}{b\rho g}\cos (\theta -\frac{\alpha }{2})$
4. $\frac{2S}{b\rho g}\cos (\theta +\frac{\alpha }{2})$

Q.

A square plate of edge d and a circular disc of diameter d are placed touching each other at the midpoint of an edge of the plate as shown in figure (9-Q2).Locate the centre of mass of the combination, assuming same mass per unit area for the two plates.

Q.

A block of mass $M$ with a semicircular groove of radius $R$ rests on a horizontal frictionless surface. A uniform cylinder of radius $r$ and mass $m$ is released from rest from the top point $A$. The cylinder slips on the semicircular frictionless track. Then find the distance travelled by the block when the cylinder reaches the bottom-most point $B$.

1. $\frac{M(R-r)}{M+m}$
2. $\frac{m(R-r)}{M+m}$
3. $\frac{(M+m)R}{M}$
4. None

Q.

Consider regular polygons with number of sides $n=3,4,\mathrm{5.......}$ as shown in the figure. The center of mass of all the polygons is at height $h$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $\Delta$. Then $\Delta$ depends on $n$ and $h$ as

1. $\Delta =h\sin \left(\frac{2\pi }{n}\right)$

2. $\Delta =h{\sin }^2\left(\frac{\pi }{n}\right)$

3. $\Delta =h{\tan }^2\left(\frac{\pi }{2n}\right)$

4. $\Delta =h(\frac{1}{\cos \left(\frac{\pi }{n}\right)}-1)$

Q.

A bird moves with a velocity of $20m{s}^{-1}$ in a direction making an angle of $60°$ with the eastern line and $60°$ with vertical upward. Represent the velocity vector in rectangular form.

Q. A body $A$ of mass $M$ while falling verticallly downwards under gravity breaks into two parts: a body $B$ of mass $\frac{M}{3}$ and a body $C$ of mass $\frac{2M}{3}$. The centre of mass of bodies $B\&C$ taken together shifts compared to that of body $A$ towards

1. depends on height of breaking
2. body $C$
3. body $B$
4. does not shift

Q.

An artificial satellite releases a bomb. Neglecting air resistance, the bomb will,

1. Strick the earth under the satellite at the instant of release

2. Strick the earth under the satellite at the instant of impact

3. Strick the earth ahead of the satellite at the instant of impact

4. Never strike earth

Q.

A tunnel is dug along a diameter of the earth. Find the force on a particle of mass m placed in the tunnel at a distance x from the centre.

Q. Two positive and equal charges are fixed at certain distance. A third charge (small) is placed in between the line joining of the two charges and it experiences zero net force due to the other two.

1. The equilibrium is stable if small charge is positive
2. The equilibrium is stable if small charge is negative
3. The equilibrium is always stable
4. The equilibrium is not stable

Q. A particle is projected vertically upwards with a velocity $u$ from a point O. When it returns to the point of projection:

1. Its average velocity is zero
2. Its displacement is zero
3. Its average speed is $\frac{u}{2}$
4. Its average speed is $u$.

Q. Two blocks A & B of mass m and 2m respectively are connected by a spring of spring constant k. the masses are moving to the right with a uniform velocity v' each, the heavier mass leading the lighter one. The spring has its natural length during this motion . Block B collides heea on with a third block C of mass 2m at rest , the collision being completely elastic.
1) then tell the velocity of block B just after collision.

2) the velocity of the COM of system of blocks A , B and C. is:-

3) maximum compression of the spring after collision is:-
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Q. A wooden plank of mass $20\text{kg}$ and length $10\text{m}$ is resting on a smooth horizontal floor. A man of mass $60\text{kg}$ starts moving from one end of the plank to the other end. Find the magnitude of displacement of the plank over the floor when the man reaches the other end of the plank.

1. $7.5\text{m}$
2. $2.5\text{m}$
3. $5\text{m}$
4. $12.5\text{m}$

Q.

Find the ratio of the magnitude of the electric force to the gravitational force acting between two protons.

Q. A boy of mass $20\text{kg}$ is standing on a $80\text{kg}$ free to move long cart. There is negligible friction between cart and ground. Initially, the boy is standing $25\text{m}$ from a wall. If he walks $10\text{m}$ on the cart towards the wall, then the final distance of the boy from the wall will be

1. $12.5\text{m}$
2. $15\text{m}$
3. $15.5\text{m}$
4. $17\text{m}$

Q. A shell following a parabolic path explodes somewhere in its flight due to some internal forces. The centre of mass of fragments will move in

1. $\text{Vertical direction}$
2. $\text{Same parabolic path}$
3. $\text{Any direction}$
4. $\text{Horizontal direction}$

Q.

A cart of mass M is tied at one end of a massless rope of length 10 m. The other end of the rope is in the hands of a man of mass M. The entire system is on a smooth horizontal surface. Initially, the man is at x = 0 and the cart at x = 10 m. If the man pulls the cart using the rope, the man and the cart will meet at the point

1. x = 5 m

2. x = 0

3. They will never meet.

4. x = 10 m

Q. Force acting on a particle is $10\text{N}$. If units of length and time are doubled and the unit of mass is halved then find the numerical value of force in the new system of units.

1. $2.5$
2. $3.5$
3. $4.5$
4. $5.5$

Q. Two blocks $A$ and $B$ are connected by a massless string (shown in the figure). A force of $30\text{N}$ is applied to block $B$. The distance travelled by centre of mass of the system in $2\text{s}$ starting from rest is

1. None of these
2. $1\text{m}$
3. $2\text{m}$
4. $3\text{m}$

Q. A body of mass 5 kg is projected up an inclined plane at 30 degree with the horizontal with an initial velocity of 6 metre per second if the frictional force opposing is motion 4.5 Newton then calculate the distance travelled by the body before coming to rest.

Q. A man of mass $M=50\text{kg}$ stands at one end of a boat (of length $L$) which is floating in still water. The man walks to the other end of the boat. Length of the boat is $L=10\text{m}$ and mass $M=20\text{kg}$. Then the distance moved by the boat relative to the ground is

1. $\frac{50}{7}\text{m}$
2. $7.5\text{m}$
3. $\frac{25}{7}\text{m}$
4. $\frac{12.5}{7}\text{m}$

Q. A man can throw a stone 80 m. The maximum height to which he can raise the stone is

1. 15 m
2. 30 m
3. 40 m
4. 10 m