The force between two atoms in a diatomic molecule can be represented approximately by the potential energy function U = U 0 a x 12 - 2 a x 6 , where U 0 and a are constants. (a) At what value of x is the potential energy zero? (b) Find the force F x (c) At what value of x is the potential energy a minimum?

Given that Potential energy, U = U 0 a x 12 - 2 a x 6 (a) If the potential energy is zero ⇒ 0 = U 0 a x 12 - 2 a x 6 ⇒ a x 12 = 2 a x 6 We get, x = a 2 1 6 (b) The force is negative derivative of potential energy function. F x = - d U d x ⇒ F x = - d d x U 0 a x 12 - 2 a x 6 ⇒ F x = 12 U 0 a 12 1 x 13 - a 6 1 x 7 ⋯ i (c) The potential energy has its minimum value when d U d x = 0 ⇒ 12 U 0 a 12 1 x 13 - a 6 1 x 7 = 0 ⇒ a 12 1 x 13 = a 6 1 x 7 We get, x = a . The minimum occurs at x = a , which is the average spacing between atoms in such a molecule.

The force between two atoms in a diatomic molecule can be represented approximately by the potential energy function U = U 0 a x 12 - 2 a x 6 , where U 0 and a are constants. (a) At what value of x is the potential energy zero? (b) Find the force F x (c) At what value of x is the potential energy a minimum?

Given that Potential energy, U = U 0 a x 12 - 2 a x 6 (a) If the potential energy is zero ⇒ 0 = U 0 a x 12 - 2 a x 6 ⇒ a x 12 = 2 a x 6 We get, x = a 2 1 6 (b) The force is negative derivative of potential energy function. F x = - d U d x ⇒ F x = - d d x U 0 a x 12 - 2 a x 6 ⇒ F x = 12 U 0 a 12 1 x 13 - a 6 1 x 7 ⋯ i (c) The potential energy has its minimum value when d U d x = 0 ⇒ 12 U 0 a 12 1 x 13 - a 6 1 x 7 = 0 ⇒ a 12 1 x 13 = a 6 1 x 7 We get, x = a . The minimum occurs at x = a , which is the average spacing between atoms in such a molecule.

A conservative force has the potential energy function U x as shown in the graph. A particle moving in one dimensional under the influence of this force has kinetic energy 1 . 0 J when it is at position x 3 . (a) Find the mechanical energy of the particle. (b) Can the particle reach either x 1 or x 5 ? (c) Find the least and greatest value of x between which the particle can move.

(a) The mechanical energy of the particle, E = U + K . . . i . It is given that at position x 3 , the kinetic energy of the particle is 1 . 0 J . From given graph at position x 3 , the potential energy of the particle is - 2 . 0 J . Hence from ( i ) , E = - 2 . 0 + 1 . 0 = - 1 . 0 J (b) At position x 1 and x 5 the potential energy of the particle is zero. The particle has mechanical energy - 1 . 0 J ,which is remains constant. It means the kinetic energy of the particle at these positions should be K = E - U = - 1 . 0 - 0 = - 1 . 0 J which cannot be possible as the kinetic energy of the particle cannot be negative. It means the particle cannot reach at the positions either x 1 or x 5 . (c) At positions x 2 and x 4 , the potential energy of the particle is - 1 . 0 J . As the mechanical energy of the particle is - 1 . 0 J , it means the kinetic energy at these positions will be zero. It suggests that the particle oscillate between the points x 2 and x 4 . Hence x min = x 2 and x max = x 4 .

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The force between two atoms in a diatomic molecule can be represented approximately by the potential energy function $U = U_{0} [{(\frac{a}{x})}^{12} - 2 {(\frac{a}{x})}^{6}],$ where $U_{0}$ and $a$ are constants.

(a) At what value of $x$ is the potential energy zero?
(b) Find the force $F_{x}$
(c) At what value of $x$ is the potential energy a minimum?

Important Questions on Work, Energy and Power

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JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 8 - Work, Energy and Power>ILLUSTRATION 8.44>Q (a)

A single conservative force $F (x)$ acts on a $1.0 kg$ particle that moves along $x$ -axis. The potential energy $U (x)$ is given by, $x = a .$ where $x$ is in meters. At $x = 5.0 m,$ the particle has kinetic energy $16 J .$

(a) What is the mechanical energy of the system.
(b) What are the least and greatest values of $x$ between which the particle can move.
(c) The maximum kinetic energy of the particle and value of $x$ at which it occurs.
(d) Determine the equation for $F (x)$ as a function of $x .$

HARD

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 8 - Work, Energy and Power>ILLUSTRATION 8.45>Q (a)

A conservative force has the potential energy function $U (x)$ as shown in the graph. A particle moving in one dimensional under the influence of this force has kinetic energy $1.0 J$ when it is at position $x_{3} .$

Question Image

(a) Find the mechanical energy of the particle.
(b) Can the particle reach either $x_{1}$ or $x_{5} ?$
(c) Find the least and greatest value of $x$ between which the particle can move.

HARD

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 8 - Work, Energy and Power>ILLUSTRATION 8.46

The potential energy of a particle in a certain field has the form

U = (\frac{a}{r^{2}}) - (\frac{b}{r}),

where

a

and

b

are positive constants and

r

is the distance from the centre of the field. Find the value of

r_{0}

corresponding to the equilibrium position of the particle; examine whether this position is stable.

HARD

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 8 - Work, Energy and Power>ILLUSTRATION 8.47

A particle of mass $m = 2 kg$ is free to move along $x$ axis under influence of a conservative force. The potential energy function for the particle is given by

$U (x) = a [{(\frac{x}{b})}^{4} - 5 {(\frac{x}{b})}^{2}] J$

Where $a = 1.0 J$ and $b = 1.0 m .$ If the total mechanical energy of the particle is zero,

(a) Plot this potential energy function, identify the regions where particle may be found
(b) Calculate the maximum speed of the particle.

EASY

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 8 - Work, Energy and Power>CONCEPT APPLICATION EXERCISE 8.3>Q 1

Figure shows three paths connecting points

a

and

b

. A single force

F

does the indicated work on a particle moving along each path in the indicated direction. On the basis of this information, is force

F

conservative?

Question Image

MEDIUM

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 8 - Work, Energy and Power>CONCEPT APPLICATION EXERCISE 8.3>Q 2

The potential energy of a conservative system is given by

U = a x^{2} - b x

, where

a

and

b

are positive constants. Find the equilibrium position and discuss whether the equilibrium is stable, unstable, or neutral.

EASY

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 8 - Work, Energy and Power>CONCEPT APPLICATION EXERCISE 8.3>Q 3

A particle moves in the $x - y$ plane in figure under the influence of a friction force with magnitude $3.00 N$ and acting in the direction opposite to the particle’s displacement. Calculate the work done by the friction force on particle as it moves along the following closed paths:

(a) path $O A$ followed by the return path $A O$

(b) path $O A$ followed by $A C$ and the return path $C O$

(d) each of your three answers should be non-zero.

What is the significance of this observation?

Question Image

EASY

JEE Advanced

IMPORTANT

PHYSICS For Joint Entrance Examination JEE (Advanced) Mechanics I>Chapter 8 - Work, Energy and Power>CONCEPT APPLICATION EXERCISE 8.3>Q 4

A single conservative force acting on a particle varies as

\vec{F} = (- A x + B x^{2}) \hat{i} N,

where

A

and

B

are constants and

x

is in metre. Calculate the potential energy function

U (x)

associated with this force, taking

U = 0

x = 0

The force between two atoms in a diatomic molecule can be represented approximately by the potential energy function U=U0ax12-2ax6, where U0 and a are constants. (a) At what value of x is the potential energy zero? (b) Find the force Fx (c) At what value of x is the potential energy a minimum?

Important Questions on Work, Energy and Power

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