Conservation of Energy

Q. The bob of a simple pendulum is displaced from its equilibrium position $O$ to a position $Q$, which is at height $h$ above $O$ and the bob is then released. Assuming the mass of the bob to be $m$ and time period of oscillations to be $2\text{s}$, the tension in the string, when the bob passes through $O$ is

1. $m(g+\pi \sqrt{2gh})$
2. $m(g+\sqrt{\frac{\pi }{2}gh})$
3. $m(g+\sqrt{\frac{{\pi }^2}{3}}gh)$
4. $m(g+\sqrt{{\pi }^{2}gh})$

Q. A pendulum bob of mass $m$ connected to the end of an ideal string of length $l$ is released from horizontal position as shown in the given figure. At the lowest point, the bob makes an elastic collison with a stationary block of mass $5m$, which is kept on a frictionless surface. Choose the correct statement(s).

1. Tension in the string just after impact is $\frac{17}{9}mg$.
2. Tension in the string just after impact is $mg$.
3. The velocity of the block just after impact is $\sqrt{\frac{2gl}{3}}$.
4. The maximum height attained by the pendulum bob after impact is $\frac{4l}{9}$ (measured from the lowest position)

Q. Given below are two statements :

Statement $1$ : In a diatomic molecule , the rotational energy at a given temperature obeys Maxwell's distribution .

Statement $2$ : In a diatomic molecule , the rotational energy at a given temperature equals the translational kinetic energy for each molecule.

In the light of above statements, choose the correct answer from the options given below:

1. Both statement $1$ and statement $2$ are false .
   
2. Both statement $1$ and statement $2$ are true .
3. Statement $1$ is false but statement $2$ is true
4. Statement $1$ is true but statement $2$ is false .

Q. A sphere of radius $0.1\text{m}$ and mass $8\pi \text{kg}$ is attached to the lower end of a steel wire of length $5.0\text{m}$ and diameter ${10}^{-3}\text{m}$. The wire is suspended from $5.22\text{m}$ high celling of a room. When the sphere is made to swing as a simple pendulum, it just grazes the floor at its lowest point. Calculate the velocity of the sphere at the lowest position.

$(Y_{\text{steel}}=1.994\times {10}^{11}{\text{N/m}}^2)$

1. $4.4{\text{ms}}^{-1}$
2. $2.2{\text{ms}}^{-1}$
3. $6.6{\text{ms}}^{-1}$
4. $8.8{\text{ms}}^{-1}$

Q. The rest mass of a deuteron ${}_1{\text{H}}^2$ is equivalent to an energy of $1876\text{MeV}$, the rest mass of a proton is equivalent to $939\text{MeV}$ and that of a neutron is $940\text{MeV}$. A deuteron may disintegrate to a proton and a neutron if it

1. emits a $\gamma$-rays photon of energy $2\text{MeV}$.
2. captures a $\gamma$-rays photon of energy $2\text{MeV}$.
3. emits a $\gamma$-ray photon of energy $3\text{MeV}$.
4. captures a $\gamma$-ray photon of energy $3\text{MeV}$.

Q. The string of a pendulum is attached to a bob and is released from its initial rest position. A fixed peg is present in the path of motion of the pendulum. What are the velocities of the bob at positions $B$ and $C$ respectively? The distances are as shown in the figure.

1. $3.16\text{m/s}$ and $7.07\text{m/s}$
2. $7.07\text{m/s}$ and $3.16\text{m/s}$
3. $5.34\text{m/s}$ and $10.78\text{m/s}$
4. $7.07\text{m/s}$ and $6.32\text{m/s}$

Q. Given the masses of various atomic particles $m_p=1.0072u,$ $m_n=1.0087u,m_e=0.000548u,m_{\overline{v}}=0,m_d=2.01241u,$ where $p=\text{proton,}n=\text{neutron,}e=\text{electron,}$
$\overline{v}=\text{antineutrino and}d=\text{deuteron.}$ Which of the following process is allowed by momentum and energy conservation?

1. $n+n\to$ deuterium atom (electron bound to the nucleus)
2. $p\to n+e^{+}+\overline{v}$
3. $n+p\to d+\gamma$
4. $e^{+}+e^{-}\to \gamma$

Q. Bombardment of lithium with protons gives rise to the following reaction.
${}_3{\text{H}}^7{+}_1{\text{H}}^1\to 2{(}_2{\text{He}}^4)+Q$.
Find the $Q$ value of the reaction.

The atomic masses of lithium, proton and helium are $7.016u,1.008u$ and $4.004u$ respectively.

1. $16.904\text{MeV}$
2. $14.904\text{MeV}$
3. $12.904\text{MeV}$
4. $10.904\text{MeV}$

Q. When the temperature increases the length of the pendulum also increases and hence its time period increases. Thus it losses the time when temperature is increased.

1. True
2. False

Q. A simple pendulum consists of a bob of mass $m$ and a light string of length $l$ as shown in the figure. Another identical ball moving with a small velocity $v_0$ collides with the pendulum's bob and sticks to it. For this new pendulum of mass $2m$, which of the following statement(s) is/are correct ?

1. Time period of the pendulum is $2\pi \sqrt{\frac{l}{g}}$.
2. The equation of motion for this pendulum is $\theta =\frac{v_0}{2\sqrt{gL}}\sin \left[\sqrt{\frac{g}{l}}t\right]$.
3. The equation of motion for this pendulum is $\theta =\frac{v_0}{4\sqrt{gl}}\cos \left[\sqrt{\frac{g}{l}}t\right]$.
4. Time period of the pendulum is $2\pi \sqrt{\frac{2l}{g}}$.

Q. ${}_{94}{\text{Pu}}^{239}$ is undergoing $\alpha -$ decay according to the equation

${}_{94}{\text{Pu}}^{239}\to {}_{92}{\text{U}}^{235}+{}_2{\text{He}}^4$

The energy released in the process is mostly kinetic energy of the $\alpha -$ particle. However, a part of the energy is released as $\gamma -$ rays. What is the speed of the emitted $\alpha -$ particle if the $\gamma -$rays radiated out have energy of $0.90\text{MeV}?$

Given:
Mass of ${}_{94}{\text{Pu}}^{239}=239.05122u$
Mass of ${}_{92}{\text{U}}^{235}=235.04299u$
Mass of${}_2{\text{He}}^4=4.002602u$
$(1u{c}^2=931\text{MeV})$

1. $1.4\times {10}^7{\text{ms}}^{-1}$
2. $2.8\times {10}^7{\text{ms}}^{-1}$
3. $3.5\times {10}^7{\text{ms}}^{-1}$
4. $6\times {10}^7{\text{ms}}^{-1}$

Q. A nucleus with mass number $220$ initially at rest emits an $\alpha -$particle. If the $Q$ value of the reaction is $5.5\text{MeV}$ then the kinetic energy $\left(\text{in MeV}\right)$ of the $\alpha -$particle is

1. $6.5$
2. $5.6$
3. $4.4$
4. $5.4$

Q. A nucleus of Mass $M+\Delta m$ is at rest and decays into two daughter nuclei of equal mass $\frac{M}{2}$ each. Speed of light is $c$. The speed of daughter nuclei is

1. $c\sqrt{\frac{\Delta m}{M+\Delta m}}$
2. $c\left(\frac{\Delta m}{M+\Delta M}\right)$
3. $c\sqrt{\frac{2\Delta m}{M}}$
4. $c\sqrt{\frac{\Delta m}{M}}$

Q.
The A and B are two systems whose interaction with surroundings are as shown above, which of the following options correctly characterise the systems A and B.

1. Open system, Closed system
2. Isolated system, Closed system
3. Adiabatic system, Isolated system
4. Closed system, Isolated system

Q. In terms of time peroid of oscillations T. find the shortest time in moving a particle from $+\sqrt{A}2$ to $-\frac{\sqrt{3}}{2}A$

Q. A simple pendulum of length $1$m has a bob of mass $100$g. It is displaced through an angle of ${60}^o$ from the vertical and then released . Find out K.E. of bob when it passes through mean position.

1. $0.12$J
2. $0.24$J
3. $0.36$J
4. $0.55$J

Q. The acceleration due to gravity at a place can be determined with the help of a simple pendulum. For this purpose effective length of the pendulum is considered. If ${}^{\prime }{l}^{\prime }$ be the length of the string and ${}^{\prime }{d}^{\prime }$ the diameter of the bob then the effective length is equal to:

1. $l+\frac{d}{2}$
2. $l-d$
3. $l-\frac{d}{2}$
4. $l+d$

Q. If a simple pendulum of mass $m$ is displaced by a distance of $x$ from its mean position, then find the potential energy stored in it.

Q. A motor cycle engine delivers a power of $10\text{kW}$ by consuming petrol at the rate of $2.4\text{kg/hr}$. Find the rate of heat rejection by the exhaust, if the calorific value of petrol is $35.5\text{MJ/kg}$.
(Neglect the other losses)

1. $33.7\text{kW}$
2. $13.7\text{kW}$
3. $23.7\text{kW}$
4. $13.7\text{W}$

Q. Two pendulums of time periods $3s$ and $7s$ respectively start oscillating simultaneously from two opposite extreme positions. After how much time they will be in phase

1. $\frac{21}{2}s$
2. $\frac{21}{4}s$
3. $\frac{21}{10}s$
4. $\frac{21}{8}s$

Q. A Carnot engine absorbs 750 J of heat energy from a reservoir at 137C and rejects 500 J of heat during each cycle, then the temperature of sink is

1. 0.25C
2. 0.34C
3. 0.44C
4. 0.54C

Q. How much energy must a gamma ray photon have, if it is to materialize into a pair of electron and positron, with each particle having a $K.E$ of $1\text{MeV}$?

1. $2\text{MeV}$
2. $1.0\text{MeV}$
3. $0.5\text{MeV}$
4. $3.02\text{MeV}$

Q. A simple pendulum has time period $T_1$. The point of suspension is now moved upward according to the relation $y=K{t}^2$, ($K=1m{s}^{-2}$) 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 $g=10m{s}^{-2}$)

1. $\frac{6}{5}$
2. $\frac{5}{6}$
3. $\frac{4}{5}$
4. $\frac{1}{2}$

Q. An insulated box containing a diatomic gas of molecular mass $M$ is moving with velocity $v$. If the box is suddenly stopped, the resulting change in temperature is ($R$ is the gas constant)

1. $\frac{M{v}^2}{2R}$
2. $\frac{M{v}^2}{3R}$
3. $\frac{M{v}^2}{5R}$
4. $\frac{2M{v}^2}{5R}$

Q. In nuclear reactions, we have the conservation of

1. Mass and energy only
2. Momentum only
3. Energy only
4. Charge, total energy and momemtum

Q. Describe the experiment simple pendulum to find the value of acceleration due to gravity.

Q. A seconds pendulum is suspended in a car that is traveling with a constant speed of $10m/s$ round a circle of radius $10m$. If the pendulum undergoes small oscillations, the time period will be

1. $2s$
2. $4s$
3. greater than $2s$ but less than $4s$
4. less than $2s$

Q. The Steam engine is based on which scientific principle?

1. Laws of thermodynamics
2. Faradays laws of electromagnetic induction
3. Ampere's Law
4. Total internal reflection of light

Q. define thermodynamics system ? is cylinder is system ? or surrounding ? pleasr explain

Q. A clock S is based on oscillations of a spring and a clock P is based on pendulum motion. Both clock run at the same rate on earth. On a planet having the same density as earth, but twice the radius:

1. S will run faster than P
2. they will both run at the equal rates, but not the same as on the earth
3. they will both run at the same rates as on the earth
4. P will run faster than S