91Ó°ÊÓ

a) Range of the distance, $0<\mathrm{x}<6\mathrm{m}$

b) Graph for $U\left(x\right)$against x is given.

c) Mechanical energy of the system, E =4J

We can find the forces at different x associated with P.E of the particle by taking the slope of the tangents at different values of x. From this, we can easily plot F(x) vs. x. Then using the law of conservation of energy, we can find the values of K.E at different x and then plot the K.E of the particle.

Formulae:

The force of a body due to potential energy,$F\left(x\right)=\frac{-dU}{dx}$ (i)

Applying law of conservation, $E=cons\tan t$(ii)

Using equation (i), we can find force at different x by taking slope of the graph $U\left(x\right)$vs.x at corresponding x. Using this, we can plot the graph of F(x) vs x. as below.

Hence, the required graph for force is plotted.

According to the law of conservation of energy, mechanical energy, which is the sum of P.E and K.E, should be constant and is equal to 4J. From this, we can plot the K.E of the particle as follows:

Hence, the required graph for potential energy is plotted.