Question

If a planet of mass M has angular momentum L
about sun, then its areal speed is given by
(2) 2M
(1)
(3) 24​

Answer 1

areal speed is given by dA/dt = L/2M.

explanation : areal speed is given as dA/dt ,

where dA → small area trace during time dt.

from law of area, A = 1/2 r² θ

[ it is derived from properties of triangle, ∆ = 1/2 absinθ , here a = b = r and θ is very small so, sinθ ≈ θ

then, ∆ = 1/2 r²θ ]

now, differentiating A with respect to time,

dA/dt = 1/2 r² dθ/dt

as we know, change in angular position with respect to time is known as angular velocity.

so, dθ/dt = ω

then, dA/dt = 1/2 r²ω.......(1)

angular momentum , L = Mr²ω

⇒L/M = r²ω......(2)

from equations (1) and (2),

dA/dt = L/2M

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Answer 2

$\huge\bold\purple{Answer:-}$

areal speed is given as dA/dt ,

where dA → small area trace during time dt.

from law of area, A = 1/2 r² θ

[ it is derived from properties of triangle, ∆ = 1/2 absinθ , here a = b = r and θ is very small so, sinθ ≈ θ

then, ∆ = 1/2 r²θ ]

now, differentiating A with respect to time,

dA/dt = 1/2 r² dθ/dt

as we know, change in angular position with respect to time is known as angular velocity.

so, dθ/dt = ω

then, dA/dt = 1/2 r²ω.......(1)

angular momentum , L = Mr²ω

⇒L/M = r²ω......(2)

from equations (1) and (2),

dA/dt = L/2M