Law of Conservation of Mechanical Energy

Q.

A body of mass $1kg$ falls freely from a height of $100m$ on a platform of mass$3kg$ which is mounted on a spring having spring constant $k=1.25\times {10}^{6}N/m$. The body sticks to the platform and the springs maximum compression is found to be$x$. Given that $g=10m{s}^{–2}$, the value of $x$will be close to:

1. $4cm$

2. $8cm$

3. $80cm$

4. $40cm$

Q.

A small bob tied at one end of a thin string of length $1m$ is describing a vertical circle so that the maximum and minimum tension in the string are in the ratio$5:1$. The velocity of the bob at the highest position is ______ $m/s$. (take$g=10m/s^2$)

Q.

A ball is dropped from rest from a height of 12 m. If the ball loses 25% of its K.E. on striking the ground, what is the height to which it bounces? How do you account for the loss in K.E.

Q. Two blocks A and B of mass $m$ and $2m$ respectively are connected by a massless spring of force constant $k$. They are placed on a smooth horizontal plane. They are stretched by an amount $x$ and then released. The relative velocity of the blocks when the spring comes to its natural length is

1. $x\sqrt{\frac{3k}{2m}}$
2. $x\sqrt{\frac{2k}{3m}}$
3. $x\sqrt{\frac{2k}{m}}$
4. $x\sqrt{\frac{3k}{m}}$

Q. A small block of mass $m$ is pushed on a smooth track from position $A$ with a velocity $\frac{2}{\sqrt{5}}$ times the minimum velocity required to reach point $D$. When the block reaches point $B$, what is the direction (in terms of angle with horizontal) of acceleration of the block ?

1. ${\tan }^{-1}\left(\frac{1}{2}\right)$
2. ${\tan }^{-1}\left(2\right)$
3. ${\sin }^{-1}\left(\frac{2}{3}\right)$
4. The block never reaches point $B$

Q. A body of mass $2\text{kg}$ is thrown vertically up with kinetic energy of $500\text{J}$. What will be height at which the kinetic energy of the body becomes ${\frac{2}{3}}^{rd}$ of original value? $(g=10{\text{m/s}}^2)$

1. $6\text{m}$
2. $9.86\text{m}$
3. $8.33\text{m}$
4. $6.23\text{m}$

Q. A block slides along a track from one level to another level, by moving through an intermediate valley. Find the maximum value of $h$ (height) that the block can reach, if the initial speed of the block is $v_0$. (Assume track is frictionless)

1. $2g(d+\frac{v_0^2}{2g})$
2. $\frac{1}{2g}(2gd+v_0^2)$
3. $\frac{v_o^2}{2g}$
4. $2gd$

Q.

Does a Compressed Spring have more Mass?

Q.

A stationary football is kicked by a player such that it attains a speed of $3\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ just after the kick. Find the work done by the player on the ball if the mass of the ball is $100g$.

Q.

A person is pulling a mass m from ground on a rough hemispherical fixed surface up to the top of the hemisphere with the help of a light inextensible string as shown in the figure. The radius of the hemisphere is $R$. Find the work done by the tension in the string. Assume that the mass is moved with a negligible constant velocity, $u$. (The coefficient of friction is $\mu$)

1. $(R-1)Mg\mu$

2. $(1+R)Mg\mu$

3. $mgR(1-\mu )$

4. $MgR(1+\mu )$

Q.

The work done in raising a mass of 15 gm from the ground to a table of 1m height is

1. 15 J

2. 152 J

3. 1500 J

4. 0.15 J

Q.

A block of mass M slides along the sides of a bowl as shown in the figure. The walls of the bowl are frictionless and the base has coefficient of friction 0.2. If the block is released from the top of the side, which is 1.5 m high, where will the block come to rest ? Given that the length of the base is 15 m

1. 1 m from P

2. Mid point

3. 2 m from P

4. At Q

Q. The bob of a pendulum initially at rest is given a sharp hit to impart a horizontal velocity of $\sqrt{3.5gl}\text{m/s}$ where $l$ is the length of the pendulum in metres. Find the height (in $\text{metres}$) from the bottom, at which tension in the string becomes zero. Take $g=10{\text{m/s}}^2$.

1. $l$
2. $\frac{3l}{2}$
3. $2l$
4. Tension in the string will never be zero.

Q. A ball is dropped along smooth inclined surfaces $AB,AC$ and $AD$. The velocity acquired at $B,C,D$ be $v_B,v_C$ and $v_D$ respectively. Then

1. $v_B>v_C>v_D$
2. $v_B<v_C<v_D$
3. $v_B>v_C<v_D$
4. $v_B=v_C=v_D$

Q.

A child builds a tower from three blocks. The blocks are uniform cubes of side $2cm$ . The blocks are initially all lying on the same horizontal surface and each block has a mass of $0.1kg$.the work done by the child is

1. $4J$

2. $0.04J$

3. $6J$

4. $0.06J$

Q. In given figure, all surfaces are smooth and frictionless. Spring is compressed by $2\text{m}$ and then released. Maximum distance travelled by the block over the inclined plane is
(Take $g=10{\text{m/s}}^2$)

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

Q. A block with initial velocity $2\text{m/s}$ is sliding on a horizontal frictionless surface as shown in the figure. What is the value of $\theta$ when the block leaves the surface?

1. $\theta ={\text{cos}}^{-1}\left(\frac{19}{25}\right)$
2. $\theta ={\text{cos}}^{-1}\left(\frac{11}{25}\right)$
3. $\theta ={\text{cos}}^{-1}\left(\frac{13}{25}\right)$
4. $\theta ={\text{cos}}^{-1}\left(\frac{17}{25}\right)$

Q.

If $W_1,W_{2}and{W}_3$ represent the work done in moving a particle from A to B along three different paths 1, 2 and 3 respectively (as shown) in the gravitational field of a point mass m, find the correct relation

1. $W_1$ > $W_2$ > $W_3$

2. $W_1=W_2=W_3$

3. $W_1$ < $W_2$ < $W_3$

4. $W_2$ > $W_1$ > $W_3$

Q. A simple pendulum is released from $A$ as shown. If $M$ and $l$ represent the mass of the bob and length of the pendulum respectively, the gain in kinetic energy at B is

1. $\frac{Mgl}{2}$
2. $\frac{\sqrt{3}Mgl}{2}$
3. $\frac{Mgl}{\sqrt{2}}$
4. $\frac{2Mgl}{\sqrt{3}}$

Q. A pendulum bob of mass $10g$ is attached to a rod of length $2m$ and is allowed to move in the vertical circle. Find the minimum velocity of bob at the bottom of the circle so that the bob completes the circle.
(Take $g=10m/s^2$)

1. $10m/s$
2. $8.9m/s$
3. $0.89m/s$
4. $9m/s$

Q. A simple pendulum is released from $A$ as shown. If $M$ and $l$ represent the mass of the bob and length of the pendulum respectively, the gain in kinetic energy at B is

1. $\frac{Mgl}{2}$
2. $\frac{\sqrt{3}Mgl}{2}$
3. $\frac{Mgl}{\sqrt{2}}$
4. $\frac{2Mgl}{\sqrt{3}}$

Q. A uniform chain of length l and mass m overhangs a smooth table with its two third part lying on the table. The kinetic energy of the chain as it completely slips off the table is

1. $\frac{4}{9}mgl$
2. $\frac{13}{18}mgl$
3. $Zero$
4. $\frac{11}{18}mgl$

Q. A bob of mass $M$ is suspended through a massless string of length $L$. The horizontal velocity at position $A$ is sufficient enough to make it reach at point $B$. The angle at which the speed of bob is half of that at $A$, satisfies

1. $\theta =\frac{\pi }{4}$
2. $\frac{\pi }{4}<\theta <\frac{\pi }{2}$
3. $\frac{\pi }{2}<\theta <\frac{3\pi }{4}$
4. $\frac{3\pi }{4}<\theta <\pi$

Q. A body of certain mass is tied to one end of a light string of length $l$, whose other end is fixed at point $\text{O}$. The body is given a speed of $\sqrt{10gl}$ from the lowermost position $\text{A}$. Find the magnitude of change in velocity of body, while moving from position $\text{A}$ to $\text{B}$ as shown in figure.

1. $\sqrt{gl}\text{m/s}$
2. $\sqrt{7gl}\text{m/s}$
3. $\sqrt{18gl}\text{m/s}$
4. $\sqrt{9gl}\text{m/s}$

Q. The system shown in the figure is released from rest with the mass $2\text{kg}$ in contact with the ground. The pulley and spring are massless, and friction is absent everywhere. Then, find the speed of the $5\text{kg}$ block when the $2\text{kg}$ block leaves contact with the ground (force constant of the spring $k=40{\text{Nm}}^{-1}$ and $g=10{\text{ms}}^{-2}$)

1. $\sqrt{2}{\text{ms}}^{-1}$
2. $2\sqrt{2}{\text{ms}}^{-1}$
3. $2{\text{ms}}^{-1}$
4. $2\sqrt{3}{\text{ms}}^{-1}$

Q. A body initially at rest begins to slid along a frictionless track from a height $h$ (as shown in the figure) just completes a vertical circle of diameter $AB=D$. The height $h$ is equal to

1. $\frac{7}{5}D$
2. $\frac{3}{2}D$
3. $\frac{5}{4}D$
4. $D$

Q. A body of mass $2\text{kg}$ is attached to a light rod of length $5\text{m}$ and is allowed to move in a vertical circle. The body is given an initial velocity of $u\text{m/s}$ at the bottom-most point, such that the final velocity at the top-most point is zero. The value of $u$ is

1. $5\sqrt{2}m/s$
2. $10m/s$
3. $10\sqrt{2}m/s$
4. $20\sqrt{2}m/s$

Q. A smooth chain $AB$ of mass $m$ rests against a surface in the form of a quarter of a circle of radius $R$. If it is released from rest, the velocity of the chain after it comes over the horizontal part of the surface is

1. $\sqrt{2gR}$
2. $\sqrt{gR}$
3. $\sqrt{2gR(1-\frac{2}{\pi })}$
4. $\sqrt{2gR(2-\pi )}$

Q. A block of mass m initially at rest is dropped from a height h on to a spring of force constant k. the maximum compression in the spring is x then

1. $mgh=\frac{1}{2}k{x}^2$
2. $mg(h+x)=\frac{1}{2}k{x}^2$
3. $mgh=\frac{1}{2}k(x+h{)}^2$
4. $mg(h+x)=\frac{1}{2}k(x+h{)}^2$

Q. A heavy particles is suspended by a string of length l from a fixed point O. The particle is given a horizontal velocity $v_0$ the string slack at some angle and the particle proceeds on a parabola. Find the value of $v_0$ if the particle passes through the point of suspension.

1. $v={\lfloor gl(2+\sqrt{3})\rfloor }^{1/2}$
2. $v={\lfloor 2gl(2+\sqrt{3})\rfloor }^{1/2}$
3. $v={\lfloor gl(1+\sqrt{3})\rfloor }^{1/2}$
4. $v={\lfloor gl(2+\sqrt{3})\rfloor }^{1/3}$