Potential Energy Formulae

Q. A body is executing simple harmonic motion. At a displacement $x$, its potential energy is $E_1$ and at a displacement $y$, its potential energy is $E_2$. The potential energy $E$ at a displacement $(x+y)$ is

(Assume that potential energy at mean position is zero)

1. $\sqrt{E_1^2+E_2^2}$
2. $E_1+E_2+2\sqrt{E_1{E}_2}$
3. $\sqrt{E_1{E}_2}$
4. $E_1+E_2$

Q.

Block A of mass m is performing SHM of amplitude a. Another block B of mass m is gently placed on A when it passes through mean position and B sticks to A. Find the amplitude of new S.H.M.

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

Q. The maximum $\text{K.E.}$ of a oscillating spring is $5\text{J}$ and its amplitude $10\text{cm}$. The force constant of the spring is:

1. $100\text{Newton/m}$
2. $1000\text{Newton-m}$
3. $1000\text{Newton/m}$
4. $1000\text{watts}$

Q.

If it takes 5 minutes to fill a 15 litre bucket from a water tap of diameter $\frac{2}{\sqrt{\pi }}$ cm then the Reynolds number for the flow is (density of water = ${10}^{3}kg/m^3$ and viscosity of water = ${10}^{-3}$ Pa.s) approximately

1. 5500
2. 1100
3. 11, 000
4. 550

Q. A particle starts SHM from the mean position. Its amplitude is $A$ and total energy $E$. At an instant its kinetic energy is $\frac{3E}{4}$. Its displacement at this instant is

1. $\frac{A}{\sqrt{2}}$
2. $\frac{A}{2}$
3. $\frac{\sqrt{3}A}{2}$
4. Zero

Q. A particle moving along x-axis is being acted upon by one dimensional conservative force F. In the f - x curve shown, four points J, K, L, M are marked on the curve. Column II gives different type of equilibriums for the particle at different positions. Column I gives certain positions on the force position graph. Match the position in Column - I with the corresponding nature of equilibriums in Column - II.

$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{(A) Point J is position of}& \text{(p) Neutral equilibrium}\\ \text{(B) Point K is position of}& \text{(q) Unstable equilibrium}\\ \text{(C) Point L is position of}& \text{(r) Stable equilibrium}\\ \text{(D) Point M is position of}& \text{(s) No equilibrium}\end{array}$

1. $A\to q;B\to s;C\to r;D\to p$
2. $A\to r;B\to q;C\to s;D\to p$
3. $A\to s;B\to q;C\to r;D\to p$
4. $A\to s;B\to p;C\to r;D\to q$

Q. The variation of potential energy of a harmonic oscillator is as shown in figure. The spring constant $k$ is

1. $1\times {10}^2\text{N/m}$
2. $1.5\times {10}^2\text{N/m}$
3. $2\times {10}^2\text{N/m}$
4. $3\times {10}^2\text{N/m}$

Q.

A particle oscillates about the equilibrium position $x_0$ subject to a force that has an associated potential energy U(x). Which of the following statements about U(x) is/are true:

1. u(x) must be symmetric about x₀

2. u(x) must have a minimum at x₀

3. u(x) must be positive in the vicinity of x₀

4. u(x) may have a minimum at x₀

Q. A linear harmonic oscillator of force constant $2\times {10}^6\text{N/m}$ and amplitude $0.01\text{m}$ has a total mechanical energy of $160$ Joules. Which of the following is true?

1. $\text{Maximum K.E. is}160\text{J}$
2. $\text{Maximum potential energy is}100\text{J}$
3. $\text{Maximum P.E. is}160\text{J}$
4. $\text{Minimum P.E. is zero}$

Q. A linear simple harmonic oscillation has a total energy of $250\text{J}$. If the potential energy at mean position is $50\text{J}$, then find the maximum kinetic energy.

1. $50\text{J}$
2. $200\text{J}$
3. $100\text{J}$
4. $150\text{J}$

Q. The potential energy of a particle executing simple harmonic motion at a distance x from the equilibrium position is proportional to

1. 
2. 
3. 
4. 

Q. A linear simple harmonic oscillation has a total mechanical energy of $300\text{J}$. Its potential energy at mean position is $100\text{J}$. Find the minimum potential energy.

1. $50\text{J}$
2. $250\text{J}$
3. $100\text{J}$
4. $200\text{J}$

Q. A particle of mass $2\text{kg}$ executes simple harmonic motion and its potential energy $U$ varies with position $x$ as shown below. The period of oscillation of the particle is

1. $\frac{2\pi }{5}\text{s}$
2. $\frac{2\sqrt{2}\pi }{5}\text{s}$
3. $\frac{\sqrt{2}\pi }{5}\text{s}$
4. $\frac{4\pi }{5}\text{s}$

Q. An oscillating mass spring system has a mechanical energy 1 joule, when it has an amplitude 0.1m and maximum speed of 1$m{s}^{-1}$. Then the force constant of the spring is

1. $100N.m^{-1}$
2. $200N.m^{-1}$
3. $50N.m^{-1}$
4. $500N.m^{-1}$

Q. In the equation $x=A\sin (\omega t+\frac{\pi }{4})$. Match the following, when $x=\frac{A}{2}$

$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{(A) Kinetic energy}& \text{(p) Half the maximum value}\\ \text{(B) Potential energy}& \text{(q) 3/4 times the maximum value}\\ \text{(C) Acceleration}& \text{(r) 1/4 times the maximum value}\\ & \text{(s) Cannot say anything}\end{array}$

1. $A\to q,B\to s,C\to p$
2. $A\to q,B\to p,C\to s$
3. $A\to q,B\to p,C\to s$
4. $A\to q,B\to s,C\to s$

Q. The potential energy of the spring increases times, when the elongation of the spring is doubled.

1. two
2. three
3. four

Q. If a spring of stiffness ${}^{\prime }{K}^{\prime }$ is cut into two parts ${}^{\prime }{A}^{\prime }$ and ${}^{\prime }{B}^{\prime }$ of length $l_A=l_B=2:3$, then the stiffness of spring ${}^{\prime }{A}^{\prime }$ is given by:

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

Q. Two similar springs $P$ and $Q$ have spring constants $k_P$ and $k_Q$, such that $k_P>k_Q$. They are stretched, first by the same amount (case a ), then by the same force (case b ). The work done by the springs $W_P$ and $W_Q$ are related as, in case (a) and case (b), respectively

1. $W_P=W_Q;W_P>W_Q$
2. $W_P=W_Q;W_P=W_Q$
3. $W_P>W_Q;W_Q>W_P$
4. $W_P<W_Q;W_Q<W_P$

Q. Two similar springs $P$ and $Q$ have spring constants $k_P$ and $k_Q$, such that $k_P>k_Q$. They are stretched, first by the same amount (case a ), then by the same force (case b ). The work done by the springs $W_P$ and $W_Q$ are related as, in case (a) and case (b), respectively

1. $W_P=W_Q;W_P>W_Q$
2. $W_P=W_Q;W_P=W_Q$
3. $W_P<W_Q;W_Q<W_P$
4. $W_P>W_Q;W_Q>W_P$

Q.

During a time-interval of exactly one period of vibration of a tuning fork, the emitted sound travels a distance

1. equal to the length of tuning fork

2. of about $330m$

3. which decrease with time

4. of one wavelength in air

Q. The variation of PE of harmonic oscillator is as shown in figure. The spring constant is

1. $1\times {10}^2{\text{Nm}}^{-1}$
2. $1.5\times {10}^2{\text{Nm}}^{-1}$
3. $2\times {10}^2{\text{Nm}}^{-1}$
4. $3\times {10}^2{\text{Nm}}^{-1}$

Q.

A force shown in the $F-x$ graph is applied to a $5kg$ cart, which then climbs up a ramp as shown. The maximum height $Y_{max}$ the cart can reach (in $m$) is

Q.

Plancks constant has the dimensions as that of

1. Power

2. Energy

3. Angular Momentum

4. Linear Momentum

Q. $3$ identical masses are released from rest. What will be their velocities when the separation between any $2$ masses becomes $a/2$ ?

1. $\sqrt{\frac{Gm}{a}}$
2. $\sqrt{\frac{2Gm}{a}}$
3. $\sqrt{\frac{Gm}{2a}}$
4. $\sqrt{\frac{2Gm}{3a}}$

Q. In the equation $x=A\sin (\omega t+\frac{\pi }{4})$. Match the following, when $x=\frac{A}{2}$

$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{(A) Kinetic energy}& \text{(p) Half the maximum value}\\ \text{(B) Potential energy}& \text{(q) 3/4 times the maximum value}\\ \text{(C) Acceleration}& \text{(r) 1/4 times the maximum value}\\ & \text{(s) Cannot say anything}\end{array}$

1. $A\to q,B\to p,C\to s$
2. $A\to q,B\to s,C\to p$
3. $A\to q,B\to p,C\to s$
4. $A\to q,B\to s,C\to s$

Q.

Why should Gibbs free energy be negative?

Q. Assertion : In simple harmonic total mechanical energy can be negative also
Reason : Potential energy is always negative and if it is greater than kinetic energy total mechanical energy will be negative.

1. Both assertion and reason are true but reason is not the correct explanation of assertion.
2. Both assertion and reason are true and reason is the correct explanation of assertion.
3. Assertion is true but reason is not true.
4. Both assertion and reason are not true.

Q. If Surface energy=Surface tension Then how are we getting the formula surface tension= surface energy/surface area?

Q. A particle moves according to $x=Acos\left(\frac{\pi }{2}t\right)$, the distance covered by it in time interval t = 0 to t = 6 s is

1. 2A
2. 3A
3. 4A
4. A

Q. For a particle executing simple harmonic motion, the kinetic energy K is given by :

1. $K_0$
2. zero
3. $K_0/2$
4. not obtainable