A particle of mass m describes the logarithmic spiral r=r0 e^b θ under a central force 𝐅 directe

Name: Problem 95, Chapter 12: Vector Mechanics For Engineers: Dynamics
Uploaded: 2020-07-27T04:20:27-08:00
Duration: 3 min 18 s
Channel: Deepak Kohli
Description: A particle of mass m describes the logarithmic spiral r=r0 e^b θ under a central force 𝐅 directed toward the center of force O. Using Eq. (12.35), show that 𝐅 is inversely proportional to the cube of the distance r from the particle to O.

step 1 Hello students, let's start our discussion. Suppose a particle of mass M, okay, its logarithm sp

Question

A particle of mass $m$ describes the logarithmic spiral $r=r_0 e^{b \theta}$ under a central force $\mathbf{F}$ directed toward the center of force $O$. Using Eq. (12.35), show that $\mathbf{F}$ is inversely proportional to the cube of the distance $r$ from the particle to $O$.

   A particle of mass $m$ describes the logarithmic spiral $r=r_0 e^{b \theta}$ under a central force $\mathbf{F}$ directed toward the center of force $O$. Using Eq. (12.35), show that $\mathbf{F}$ is inversely proportional to the cube of the distance $r$ from the particle to $O$.

Vector Mechanics For Engineers: Dynamics

Beer Ferdinand P And… 11th Edition

Chapter 12, Problem 95

Instant Answer

Solved by Expert Deepak Kohli

07/27/2020

Step 1

Equation (12.35) likely refers to the equation for central force in terms of the effective potential: $$F_r = -\frac{dV_{eff}}{dr}$$ where $V_{eff}$ is the effective potential given by: $$V_{eff}(r) = V(r) + \frac{L^2}{2mr^2}$$ with $L$ being the angular momentum. Show more…

Show all steps

Thanks for your feedback!

A particle of mass $m$ describes the logarithmic spiral $r=r_0 e^{b \theta}$ under a central force $\mathbf{F}$ directed toward the center of force $O$. Using Eq. (12.35), show that $\mathbf{F}$ is inversely proportional to the cube of the distance $r$ from the particle to $O$.