Question

Conservative force $\mathit{F}\left(x\right)$acts on a particle that moves along an x axis. Figure 8-72 shows how the potential energy $\mathit{U}\left(x\right)$associated with force $\mathit{F}\left(x\right)$varies with the position of the particle, (a) Plot $\mathit{F}\left(x\right)$for the range $\mathbf{0}\mathbf{<}\mathit{x}\mathbf{<}\mathbf{6}\mathit{m}$. (b) The mechanical energy Eof the system is 4.0J. Plot the kinetic energy localid="1661232921223" $\mathit{K}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$of the particle directly on Fig. 8-72.

Short answer

a) $F\left(x\right)$for the range $0<x<6m$is plotted.

b) Kinetic energy $K\left(x\right)$of the particle on the given figure is plotted.

Step-by-step solution

The given data

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

Understanding the concept of energy and force

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)

a) Calculation for plotting the force versus distance graph

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.

b) Calculation for plotting the potential energy versus distance graph

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.