A planet orbits its star in a circular orbit (uniform circular motion) of radius 1.62x10^11 m. The orbital period of the planet around its star is 37.0 years. Determine the following quantities for this orbital motion: Angular acceleration , Tangential acceleration, Radial acceleration, Angular velocity, and Tangential velocity
To determine the quantities for this orbital motion, we can use the following formulas:
1. Angular Acceleration (α):
Angular acceleration is the rate at which the angular velocity of an object changes over time. In uniform circular motion, the angular acceleration is zero because the angular velocity remains constant. Therefore, α = 0.
2. Tangential Acceleration (a_t):
Tangential acceleration is the acceleration along the tangent (perpendicular to the radius) of the circular path. It is given by the formula:
a_t = r * α,
where r is the radius of the circular orbit, and α is the angular acceleration.
Since α = 0, the tangential acceleration (a_t) is also zero.
3. Radial Acceleration (a_r):
Radial acceleration is the acceleration in the direction of the radius of the circular path. It can also be called centripetal acceleration. It is given by the formula:
a_r = (v^2) / r,
where v is the tangential velocity and r is the radius of the circular orbit. We'll calculate v first.
4. Angular Velocity (ω):
Angular velocity is the rate at which an object rotates around a fixed point. In uniform circular motion, the angular velocity (ω) is constant and is given by the formula:
ω = 2π / T,
where T is the period of the orbital motion (time taken to complete one revolution). We have the value of T, so we can calculate ω.
5. Tangential Velocity (v):
Tangential velocity is the linear velocity along the tangent of the circular path. It is given by the formula:
v = r * ω,
where r is the radius of the circular orbit, and ω is the angular velocity.
Let's calculate these quantities:
1. Angular Acceleration (α) = 0 (in uniform circular motion).
2. Tangential Acceleration (a_t) = 0 (since α = 0).
3. Radial Acceleration (a_r):
a_r = (v^2) / r,
where v = r * ω,
and ω = 2π / T.
Thus, a_r = (r * ω^2) / r = ω^2.
Substitute ω = 2π / T,
a_r = (2π / T)^2.
Now let's determine ω and v:
4. Angular Velocity (ω) = 2π / T = 2π / 37 years.
5. Tangential Velocity (v) = r * ω.
Substituting the given values into the formulas, we can find the numerical values for each quantity.