Chapter 8, Problem 8.1CYU

Check Your understanding In Example 8.1 what are the potential energies of the particle at $x=1$ m and $x=2$ m with respect to zero at $x=1.5$ ?verify that the difference of potential energy is still 7 J.

To determine

The potential energies of the particle at $x=1\text{m}$ and $x=2\text{m}$ with respect to zero at $x=1.5\text{m}$ and to check whether the difference of potential energy is $7\text{J}$ or not.

Answer to Problem 8.1CYU

The potential energies of the particle at $x=1\text{m}$ and $x=2\text{m}$ are $-2.375\text{J}$ and $4.625\text{J}$ respectively and the difference between the potential energy at $x=1$ and $x=2$ is still $7\text{J}$.

Explanation of Solution

Given:

The potential energy at $x=1.5$ is $0$.

Formula used:

The potential energy is given by,

$U\left(x\right)=\frac{1}{3}a{x}^3+C$

Here, $C$ is a constant.

Calculation:

The potential energy is calculated as,

$\begin{array}{c}U\left(x\right)=\frac{1}{3}a{x}^3+C\\ =\frac{1}{3}\left(3\right)x^3+C\\ =x^3+C\end{array}$

At $x=1.5$ ,

$U\left(1.5\right)=0\text{J}$

The value of the constant is calculated as,

$\begin{array}{c}U\left(x\right)=x^3+C\\ U\left(1.5\right)={\left(1.5\right)}^3+C\\ 0\text{J}=3.375\text{J}+C\\ C=-3.375\text{J}\end{array}$

Therefore, the expression for potential energy is,

$U\left(x\right)=x^3-3.375$

For $x=1$ ,

The potential energy is calculated as,

$\begin{array}{c}U\left(x\right)=x^3-3.375\\ U\left(1\right)={\left(1\right)}^3-3.375\\ =-2.375\text{J}\end{array}$

For $x=2$ ,

The potential energy is calculated as,

$\begin{array}{c}U\left(x\right)=x^3-3.375\\ U\left(2\right)={\left(2\right)}^3-3.375\\ =4.625\text{J}\end{array}$

The difference between the potential energy is calculated as,

$\begin{array}{c}D=U\left(2\right)-U\left(1\right)\\ =4.625\text{J}-\left(-2.375\text{J}\right)\\ =4.625\text{J}+2.375\text{J}\\ =7\text{J}\end{array}$

Conclusion:

The potential energies of the particle at $x=1\text{m}$ and $x=2\text{m}$ are $-2.375\text{J}$ and $4.625\text{J}$ respectively and the difference between the potential energy at $x=1$ and $x=2$ is still $7\text{J}$.