Use the angular component of acceleration in polar coordinates to deduce Keplerís second law r^2 (dθ/dt)= k for a central force field.
To deduce Kepler's second law using the angular component of acceleration in polar coordinates, we need to consider the motion of a particle under the influence of a central force field.
First, let's define some variables:
- r is the radial distance of the particle from the origin.
- θ is the angle between the radial vector and a reference direction (usually the x-axis).
- t is the time.
- k is a constant.
The acceleration of the particle in polar coordinates can be expressed as:
a = (d^2r / dt^2 - r(dθ / dt)^2) r̂ + (2(dr/dt)(dθ/dt) + r(d^2θ/dt^2)) θ̂
Here, r̂ and θ̂ are the unit vectors in the radial and tangential directions, respectively.
In a central force field, the force acting on the particle is radially directed towards or away from the origin and has no tangential component. Thus, the tangential component of the acceleration (2(dr/dt)(dθ/dt) + r(d^2θ/dt^2)) θ̂ = 0.
Therefore, we have:
2(dr/dt)(dθ/dt) + r(d^2θ/dt^2) = 0
This equation simplifies to:
dθ/dt = -2(dr/dt) / r
Now, we integrate both sides of this equation with respect to time:
∫dθ = ∫(-2 / r)(dr/dt) dt
The left-hand side becomes θ, and the right-hand side becomes -2∫(1/r)(dr/dt) dt.
Simplifying further, we have:
θ = -2∫(1/r)(dr/dt) dt
Integrating both the sides, we get:
θ = -2 ∫dt
θ = -2t + C
Where C is an integration constant.
Kepler's second law states that the line joining a planet to the Sun sweeps out equal areas in equal time intervals.
According to our deduced equation for θ, we can see that θ is linearly related to time t. This implies that the rate at which the line connecting the particle to the central force sweeps out areas is constant. Therefore, the law states that r^2(dθ/dt) = k, where k is a constant.
This equation is Kepler's second law, which describes the conservation of angular momentum for a particle under the influence of a central force field. The equation shows that the rate at which the area is swept out by the radial vector is constant, regardless of the radial distance r.