Question

A single conservative force F = (7.0x - 11) N, where x is in meters, acts on a particle moving along an x axis. The potential energy U associated with this force is assigned a value of 26 J at x = 0. (a) What is the maximum positive potential energy? At what (b) negative value and (c) positive value of x is the potential energy equal to zero?

Answer 1

Answer:

(a) 34.6429J

(b) -1.57 m

(c) 4.71 m

Explanation:

The derivative of the potential energy with respect to the position is equal to the negative of the force, so:

$-\frac{dU}{dx}=F\\ dU=-Fdx$

Then, if we integer both sides, we get:

∫dU = -∫(7x - 11)dx

$U=\frac{-7}{2}x^{2} + 11x + c$

we know that U is equal to 26 J when x is zero, so:

$U=\frac{-7}{2}x^{2} + 11x + c$

$26=\frac{-7}{2}(0)^{2} + 11(0) + c$

$26=c$

Finally, the equation for the potential energy is:

$U(x)=\frac{-7}{2}x^{2} + 11x + 26$

Therefore, the maximum positive potential energy is the energy when x is equal to 11/7. That is because the equation of U is the equation of a parable, and the vertex in a parable is given by:

$x=\frac{-b}{2a} = \frac{-11}{2(-7/2)} =\frac{11}{7}$

Where b is the number beside x and a is the number beside $x^{2}$, Then, the value of maximum U is:

$U(11/7)=\frac{-7}{2}(11/7)^{2} + 11(11/7) + 26$

$U(11/7)=34.6429J$

On the other hand, the negative and positive values of x where the potential energy is equal to zero is calculated as:

$U(x)=\frac{-7}{2}x^{2} + 11x + 26$

$0=\frac{-7}{2}x^{2} + 11x + 26$

if we solve this using the quadratic equation, we get:

$x =\frac{-11+\sqrt{11^{2}-4(-7/2)(26)} }{2(-7/2)}=-1.5747$

$x =\frac{-11-\sqrt{11^{2}-4(-7/2)(26)} }{2(-7/2)}=4.7175$

Finally, the negative and positive values of x where the potential energy is equal to zero are -1.5747 and 4.7175 respectively.

Answer 2

The negative value and positive value of x where the potential energy is equal to zero are -1.5747 and 4.7175 respectively.

What is potential energy?

Potential energy is the energy associated with the position of an object.

We know that the force in terms of potential energy with respect to the position of the particle can be written as,

$-F = \dfrac{dU}{dx}$,

where dU is the change in the potential energy of the particle for a distance traveled dx.

Now, the potential energy from the equation of the force can be written as,

$-F\ dx = dU\\dU = -F\ dx$

As the function of the force in terms of position is given to us, substitute the function and then integrate both the sides of the equation,

$dU = -(7x-11)dx\\\int dU = \int -7x+11\ dx\\U = -7\dfrac{x^2}{2}+11x+C$

Where C is a constant.

As the potential energy U is associated with this force, when the particle is at its initial condition, therefore, x=0 is given as 26 J, substitute in order to find the value of the constant C,

$U = -7\dfrac{x^2}{2} + 11x+C\\\\26 = -7\dfrac{(0)^2}{2} + 11(0)+C\\\\C = 26$

Thus, the function of U gives us the potential energy within the particle at position x is $U = -7\dfrac{x^2}{2}+11x+26$.

B.) The maximum potential energy with the object can be found using the function.

Now, since the equation of the potential energy for this particle is the equation of a parabola, therefore, the maximum potential energy of the particle will lie at its vertex of the function. The value of x at the vertex of the parabola is,

$x = \dfrac{-b}{2a}=\dfrac{-11}{2\times \frac{-7}{2}} = \dfrac{11}{7}$

As the vertex of the parabola lies at $\dfrac{11}{7}$, therefore,

The maximum potential energy of the particle is,

$U = -7\dfrac{(\frac{11}{7})^2}{2}+11(\frac{11}{7})+26$

U = 34.6429 J

C.) In order to calculate the negative value and positive value of x where the potential energy is equal to zero we will calculate the factors of the quadratic equation of the parabola. Therefore, the function can be written as,

$U = -7\dfrac{x^2}{2}+11x+26$

As the value of the potential energy is 0,

$0 = -7\dfrac{x^2}{2}+11x+26$

Substitute the values in the formula of the quadratic equation, we will get the values of x as,

x = -1.5747

x= 4.7175

Hence, the negative value and positive value of x where the potential energy is equal to zero are -1.5747 and 4.7175 respectively.

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