Question

a) Write the equation of movement for a particle in a central field given by $f(r) \hat{r}$ b) Prove that the angular part $r^2 \dot{\theta} = \text{constant} = L$. c) Prove that by substituting the differential equation for the trajectory of a particle in a central field is: $\frac{d^2u}{d\theta^2} + u = -\frac{f(1/u)}{mL^2u^2}$ d) Prove that if a central force is defined by: $f(r) = -k/r^2$ such that $k>0$, then the trajectory of the particle is conic

          a) Write the equation of movement for a particle in a central field given by $f(r) \hat{r}$ 
b) Prove that the angular part $r^2 \dot{\theta} = \text{constant} = L$.
c) Prove that by substituting the differential equation for the trajectory of a particle in a central
field is: 
$\frac{d^2u}{d\theta^2} + u = -\frac{f(1/u)}{mL^2u^2}$
d) Prove that if a central force is defined by: $f(r) = -k/r^2$ such that $k>0$, then the trajectory of
the particle is conic

a) Write the equation of movement for a particle in a central field given by f(r) r̂
b) Prove that the angular part r^2 θ̇ = constant = L.
c) Prove that by substituting the differential equation for the trajectory of a particle in a central
field is:
(d^2u)/(dθ^2) + u = -(f(1/u))/(mL^2u^2)
d) Prove that if a central force is defined by: f(r) = -k/r^2 such that k>0, then the trajectory of
the particle is conic

Added by Alisha G.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Lucas Finney

Step 1

In this case, f(r) = r, so the equation becomes: m(d^2r/dt^2) = r b) To prove that the angular part r^2 * θ' is constant, we can use the conservation of angular momentum. The angular momentum L is defined as: L = m * r^2 * θ' where θ' is the angular velocity. Show more…

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a) Write the equation of movement for a particle in a central field given by f(r) = r. b) Prove that the angular part r^2 * Î¸' = constant = L. c) Prove that by substituting the differential equation for the trajectory of a particle in a central field given by a^2 * u * f(1/u), the equation is: d^2Î¸/dt^2 + n^2 * Î¸ = -d^2Î¸/dÏ†^2 + mL^2 * u^2. d) Prove that if a central force is defined by f(r) = -k/r^2 such that k > 0, then the trajectory of the particle is conic.