Law of Conservation of Mechanical Energy

Q. Two identical charged spheres suspended from a common point by two massless strings of length $l$, are initially at a distance $d(d<<l)$ apart because of their mutual repulsion. The charges begins to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velocity $v$. Then $v$ varies as a function of the distance $x$ between the spheres as

1. $v\propto x^{\frac{1}{2}}$
2. $v\propto x$
3. $v\propto x^{-\frac{1}{2}}$
4. $v\propto x^{-1}$

Q. A bullet of mass $10g$ moving horizontally with a velocity of $400m/s$ strikes a wooden block of mass $2kg$ which is suspended by a light inextensible string of length $5m$. As a result, the centre of gravity of the block is found to rise a vertical distance of $10cm$. The speed of the bullet after it emerges out horizontally from the block will be

1. $120m/s$
2. $160m/s$
3. $100m/s$
4. $80m/s$

Q. System shown in figure is released from rest. Pulley and spring is massless and friction is absent everywhere. The speed of $5\text{kg}$ block when $2\text{kg}$ block leaves contact with ground is (Take force constant of $5\text{kg}$ spring $k=40\frac{\text{N}}{\text{m}}$ and $g=10\frac{\text{m}}{{\text{s}}^2}$

1. $2\text{m/s}$
2. $4\sqrt{2}\text{m/s}$
3. $2\sqrt{2}\text{m/s}$
4. $\sqrt{2}\text{m/s}$

Q.

What are the three equations of motion?

Q.

Which of the following property of a proton can change while it moves freely in a magnetic field? (There may be more than one correct answer.) A) mass B) speed C) velocity D) momentum

1. mass

2. speed

3. velocity

4. momentum

Q.

Two identical springs of spring constant $‘2k’$ are attached to a block of mass $m$ and to fixed support (see figure). When the mass is displaced from equilibrium position on either side, it executes simple harmonic motion. Then, time period of oscillations of this system is:

1. $\pi \sqrt{\frac{m}{k}}$

2. $\pi \sqrt{\frac{m}{2k}}$

3. $2\pi \sqrt{\frac{m}{k}}$

4. $2\pi \sqrt{\frac{m}{2k}}$

Q. What is the velocity of the bob of a simple pendulum at its mean position, if it is able to rise to a maximum vertical height of $40\text{cm}$.
(Take $g=9.8{\text{m/s}}^2$)

1. $1.4\text{m/s}$
2. $0.7\text{m/s}$
3. $2.8\text{m/s}$
4. $3.6\text{m/s}$

Q.

Obtain an expression for acceleration of the particle performing circular motion.

Q. A solid cylinder of mass $3\text{kg}$ is rolling on a horizontal surface with velocity $4\text{m/s}$. It collides with a horizontal spring of force constant $200\text{N/m}$. The maximum compression produced in the spring will be

1. $0.7\text{m}$
2. $0.2\text{m}$
3. $0.5\text{m}$
4. $0.6\text{m}$

Q. A block of mass $M$ is attached to the lower end of a vertical spring. The spring is hung from a ceiling and has force constant value $k$. The mass is released from rest with the spring initially unstretched. The maximum extension produced in the length of the spring will be

1. $2\text{Mg/k}$
2. $4\text{Mg/k}$
3. $\text{Mg/2k}$
4. $\text{Mg/k}$

Q. Two identical blocks A and B, each of mass 'm' resting on smooth floor are connected by a light spring of natural length L and spring constant K, with the spring at its natural length. A third identical block 'C' (mass m) moving with a speed v along the line joining A and B collides with A. the maximum compression in the spring is

1. 
2. 
3. 
4. 

Q. If the gravitational force were proportional to $\frac{1}{r}$, then a particle in a circular orbit under such a force would have its original speed $v$
[$r$ is the radius of circular orbit]

1. Independent of $r$
2. $\propto \frac{1}{r}$
3. $\propto \frac{1}{r^2}$
4. $\propto r^2$

Q.

A cubical vessel of height $1\mathrm{m}$ is full of water. What is the work done in pumping water out of the vessel?

Q. A mass is suspended separately by two springs and the time periods in the two cases are $T_1$ and $T_2.$ If the same mass be suspended by connecting the two springs in parallel then the time period of oscillations is $T_p$ and when the two springs are connected in series then the time period of oscillations is $T_s$. Then select the correct relation(s).

1. $T_s=T_1+T_2$
   
2. $T_s^2=T_1^2+T_2^2$
   
3. $T_p^{-2}=T_1^{-2}+T_2^{-2}$
   
4. $T_p=T_1+T_2$
   

Q. A small sphere is given vertical velocity of magnitude $v_0=5{\text{ms}}^{-1}$ and it swings in a vertical plane about the end of a massless string. The angle $\theta$ with the vertical at which the string will break, given that it can withstand a maximum tension equal to twice the weight of the sphere, is

1. ${\cos }^{-1}\left(\frac{2}{3}\right)$
2. ${\cos }^{-1}\left(\frac{1}{4}\right)$
3. ${60}^{\circ }$
4. ${30}^{\circ }$

Q.

Water falls from a height o 500m what is the rise in temp of water at bottom of whole energy remains in the water

Q.

A small block of mass 100 g is pressed against a horizontal spring fixed at one end to compress the spring through 5.0 cm (figure 8-E11). The spring constant is 100 N/m. When released, the block moves horizontally till it leaves the spring. Where will it hit the ground 2 m below the spring ?

Q. A steel wire is suspended vertically from a rigid support. When loaded with a weight in air, it extends by and when the weight is immersed completely in water, the extension is reduced to $l_w$.Then the relative density of the weight is

1. $\frac{l_a}{l_a-l_w}$
   
   
2. $\frac{l_a}{l_w}$
   
3. $\frac{l_2}{l_a-l_w}$
   
4. $\frac{l_w}{l_a}$
   

Q. A ball of mass $m$ is released from rest at $\text{A}$ as shown in the figure. Find the minimum height $h$, so that ball will just complete the circular motion on the track (all surfaces are smooth).

1. $h=2R$
2. $h=4R$
3. $h=R$
4. $h=\frac{5}{2}R$

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 is sitting on a swing whose minimum and maximum heights from the ground are $0.75\text{m}$ and $2\text{m}$ respectively. Its maximum speed will be

1. $10\text{m/s}$
2. $5\text{m/s}$
3. $8\text{m/s}$
4. $15\text{m/s}$

Q. Mass of block $A$ is $m$ and that of block $B$ is $2m$. Spring constant is $k$ and there is no friction. System is released from rest with the spring unstretched. Pulley is massless. Identify the correct statement(s).

1. Maximum extension of the spring is $\frac{4mg}{k}$
2. Acceleration of block B is $\frac{g}{3}$ downwards, when extension in the spring is $\frac{mg}{k}$
3. Maximum extension in the spring is $\frac{2mg}{k}$
4. Acceleration of block $B$ is $\frac{g}{3}$ downwards, when extension in spring is $\frac{4mg}{k}$

Q.

The work done by the external force on a system equals the change in _________ energy.

Q. A block $m_1$ of mass $2.0\text{kg}$ is moving on a frictionless horizontal surface with a velocity of $1.0\text{m/s}$ towards another block $m_2$ of equal mass kept at rest as shown. The spring constant of the spring fixed at one end of $m_2$ is $100\text{N/m}$. The maximum compression of the spring is

1. $2\text{cm}$
2. $5\text{cm}$
3. $10\text{cm}$
4. $20\text{cm}$

Q. Block $A$ of mass $m$ is hanging from a vertical spring having stiffness $k$ and is at rest. Block $B$ of identical mass strikes the block $A$ with velocity $v$ and sticks to it. Then the value of $v$ for which the spring just attains natural lengths is

1. $\sqrt{\frac{5m{g}^2}{k}}$
2. $\sqrt{\frac{8m{g}^2}{k}}$
3. $\sqrt{\frac{6m{g}^2}{k}}$
4. $\sqrt{\frac{7m{g}^2}{k}}$

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 body of mass $5kg$ is thrown vertically up with a kinetic energy of $490J$. The height at which the kinetic energy of the body becomes half of the original value is

1. $12.5m$

2. $10m$

3. $2.5m$

4. $5m$

Q. A system consists of two idential cubes each of mass $m$ linked together by a massless spring of spring constant $k$. The spring is compressed by $x$ connecting cubes by thread. Find minimum value of $x$ for which lower cube will bounce up after the thread has been burnt.

1. $\frac{2mg}{k}$
2. $\frac{3mg}{k}$
3. $\frac{3mg}{2k}$
4. $\frac{mg}{2k}$

Q.

Two masses of 4 kg and 5 kg are connected by a string passing through a frictionless pulley and are kept on a frictionless table as shown in the figure. The acceleration of 5 kg mass is

1. 49 m/s²

2. 5.44 m/s²

3. 19.5 m/s²

4. 2.72 m/s²

Q.

How do you find acceleration and velocity without time?