A single conservative force acts on a 1-gram particle that moves along  x -axis. The potential energy of the particle varies with  x  as  U  =  20 + (x − 2)2 , here  U  in erg and  x  in  cm . When the particle is at  x  =  5cm , its kinetic energy is  20erg . Then which of the following(s) is/are correct?Mechanical energy of particle is 49 ergLeast and greatest value of  x  between which particle can move is  (2 − √29)cm  and  (2 + √29)cm  respectivelyMaximum kinetic energy of particle is 29 ergAt  x  =  2cm , the body is in equilibrium

To answer this question, we need to understand the concepts of potential energy, kinetic energy, and mechanical energy in physics.

1. Mechanical energy of the particle is 49 erg:

Mechanical energy is the sum of kinetic energy and potential energy. Given that the kinetic energy is 20 erg when x = 5 cm, we can find the potential energy at this point using the given function U = 20 + (x - 2)^2. Substituting x = 5 cm into this equation gives U = 20 + (5 - 2)^2 = 29 erg. Therefore, the total mechanical energy is 20 erg (kinetic) + 29 erg (potential) = 49 erg. So, this statement is correct.

2. Least and greatest value of x between which particle can move is (2 - √29) cm and (2 + √29) cm respectively:

The particle can move as long as its total mechanical energy is greater than or equal to its potential energy. Setting the mechanical energy (49 erg) equal to the potential energy (20 + (x - 2)^2) and solving for x gives the range of possible x values. However, this calculation does not yield the given range, so this statement is incorrect.

3. Maximum kinetic energy of particle is 29 erg:

The maximum kinetic energy would occur when the potential energy is at its minimum. The given function for potential energy is a parabola that opens upwards, so its minimum occurs at x = 2 cm. Substituting this into the equation gives a minimum potential energy of 20 erg. Since the total mechanical energy is conserved and is 49 erg, the maximum kinetic energy is 49 erg - 20 erg = 29 erg. So, this statement is correct.

4. At x = 2 cm, the body is in equilibrium:

In physics, a body is in equilibrium when the net force acting on it is zero. For a conservative force, this occurs when the derivative of the potential energy with respect to x is zero. The derivative of U = 20 + (x - 2)^2 with respect to x is 2*(x - 2), which is zero when x = 2 cm. So, this statement is correct.

To answer this question, we need to understand the concepts of potential energy, kinetic energy, and mechanical energy.

1. Mechanical energy of the particle is 49 erg:

Mechanical energy is the sum of kinetic energy and potential energy. Given that the kinetic energy is 20 erg and the potential energy is U = 20 + (x - 2)^2, we can substitute x = 5 cm into the equation to find the potential energy at this point.

U = 20 + (5 - 2)^2 = 20 + 9 = 29 erg

Therefore, the total mechanical energy is 20 erg (kinetic) + 29 erg (potential) = 49 erg. So, this statement is correct.

2. Least and greatest value of x between which particle can move is (2 - √29)cm and (2 + √29)cm respectively:

The particle can move as long as its mechanical energy is greater than or equal to its potential energy. Setting the mechanical energy (49 erg) equal to the potential energy gives us:

49 = 20 + (x - 2)^2

Solving this equation for x gives us x = 2 ± √29. So, this statement is also correct.

3. Maximum kinetic energy of particle is 29 erg:

The maximum kinetic energy would occur when the potential energy is at its minimum. The minimum potential energy occurs at x = 2 cm, where U = 20 + (2 - 2)^2 = 20 erg. Since the total mechanical energy is conserved and equals 49 erg, the maximum kinetic energy would be 49 - 20 = 29 erg. So, this statement is correct.

4. At x = 2 cm, the body is in equilibrium:

At equilibrium, the force on the body is zero. The force is the negative derivative of the potential energy with respect to x. The derivative of U = 20 + (x - 2)^2 is 2*(x - 2). Setting this equal to zero gives x = 2 cm. So, this statement is also correct.

Therefore, all the statements are correct.