Finding 3-D potential energies (a) Show that the uniform gravity field F = −mgk is conservative with potential energy V = mgz. (b) Show that any force field of the form F = h(r)r (a central field) is conservative with potential energy V = −H(r), where H(r)

Question

Finding 3-D potential energies

(a) Show that the uniform gravity field F = −mgk is conservative with potential energy V = mgz.

(b) Show that any force field of the form

[latex] F =h(r) \widehat{ r }[/latex]

(a central field) is conservative with potential energy V = −H(r), where H(r) is the indefinite integral of h(r). Use this result to find the potential energies of (i) the 3-D SHM field [latex] F =-\alpha r \hat{ r }[/latex], and (ii) the attractive inverse square field [latex] F =-\left(K / r^{2}\right) \widehat{ r }[/latex], where α and K are positive constants.

Accepted Answer

Since the potential energies are given, it is sufficient to evaluate − grad V in each case and show that this gives the appropriate F. Case (a) is immediate. In case (b),

[latex] \frac{\partial H(r)}{\partial x}=\frac{d H}{d r} \frac{\partial r}{\partial x}=H^{\prime}(r) \frac{x}{r}=h(r) \frac{x}{r},[/latex]

since [latex] r=\left(x^{2}+y^{2}+z^{2} ight)^{1 / 2} ext { and } H^{\prime}(r)=h(r).[/latex] Thus

[latex] -\operatorname{grad}[-H(r)]=h(r)\left(\frac{x}{r} i +\frac{y}{r} j +\frac{z}{r} k ight)=h(r) \frac{ r }{r}=h(r) \widehat{ r },[/latex]

as required.

In particular then, the potential energy of the SHM field [latex] F =-\alpha r \hat{ r } ext { is } V= \frac{1}{2} \alpha r^{2} [/latex], and the potential energy of the attractive inverse square field [latex] F =-\left(K / r^{2} ight) \widehat{ r }[/latex] is V = −K/r.

Question 6.8: Finding 3-D potential energies (a) Show that the uniform gra...

Related Answered Questions

Does P leave the surface of the sphere? For the particular case in which u = (3gb)^1/2, show that P will leave the surface of the sphere, and find the value of θ at which it does so. ...

Verified Answer:

Small oscillations in constrained motion A particle P of mass m can slide freely along a long straight wire. P is connected to a fixed point A, which is at a distance 4a from the wire, by a light elastic cord of natural length 3a and strength α. Find the approximate ...

Verified Answer:

Complete circles How large must the initial speed be for P to perform complete circles? ...

Verified Answer:

The reaction force Find the reaction force N as a function of θ. ...

Verified Answer:

Motion in a vertical circle A fixed hollow sphere has centre O and a smooth inner surface of radius b. A particle P, which is inside the sphere, is projected horizontally with speed u from the lowest interior point (see Figure 6.5). Show that, in the subsequent motion, ...

Verified Answer:

The snowboarder A snowboarder starts from rest and descends a slope, losing 320 m of altitude in the process. What is her speed at the bottom? [Neglect all forms of resistance and take g = 10 m s^−2.] ...

Verified Answer:

Stability of equilibrium in a 3-D conservative field A particle P of mass m can move under the gravitational attraction of two particles, of equal mass M, fixed at the points (0, 0, ±a). Show that the origin O is a position of equilibrium, but that it is not stable. ...

Verified Answer:

Escape from the Moon A body is projected from the surface of the Moon with speed u in any direction. Show that the body cannot escape from the Moon if u² < 2MG/R, where M and R are the mass and radius of the Moon. [Assume that the Moon is spherically symmetric.] ...

Verified Answer:

Projectile motion A body is projected from the ground with speed u and lands on the flat roof of a building of height h. Find the speed with which the projectile lands. [Assume uniform gravity and no air resistance.] ...

Verified Answer:

Conservative or not conservative? Show that the field F1 = −2x i − 2y j − 2zk is conservative but that the field F2 = yi − x j is not. ...

Verified Answer: