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 equations of circular motion.
1. Angular Acceleration (α):
The angular acceleration (α) is the rate of change of angular velocity (ω). We can find it using the equation:
α = (2π) / T,
where T is the period of the planet's orbit (37.0 years).
Plugging in the given values:
α = (2π) / 37.0 years = 0.169 radians/year^2
2. Tangential Acceleration (at):
The tangential acceleration (at) is the acceleration of the planet along its circular path. In uniform circular motion, tangential acceleration is zero because the speed remains constant.
at = 0 m/s^2
3. Radial Acceleration (ar):
The radial acceleration (ar) is the acceleration directed towards the center of the circle. It is equal to the centripetal acceleration.
We can use the following equation to find radial acceleration:
ar = ω^2 * r,
where ω is the angular velocity and r is the radius of the circular orbit.
To find ω, we can use the formula:
ω = 2π / T,
where T is the period of the planet's orbit (37.0 years).
Plugging in the given values:
ω = 2π / 37.0 years ≈ 0.169 radians/year
Now we can calculate the radial acceleration:
ar = (0.169 radians/year)^2 * 1.62x10^11 m ≈ 4.611x10^19 m/year^2
4. Angular Velocity (ω):
The angular velocity (ω) is the rate at which the planet moves around the circle. We have already calculated ω in the previous step:
ω = 0.169 radians/year
5. Tangential Velocity (vt):
The tangential velocity (vt) is the linear speed of the planet along the circular path.
We can find vt using the equation:
vt = ω * r,
where ω is the angular velocity and r is the radius of the circular orbit.
Plugging in the given values:
vt = (0.169 radians/year) * 1.62x10^11 m ≈ 2.74x10^10 m/year
So, the quantities for this orbital motion are:
Angular acceleration (α) ≈ 0.169 radians/year^2
Tangential acceleration (at) = 0 m/s^2
Radial acceleration (ar) ≈ 4.611x10^19 m/year^2
Angular velocity (ω) ≈ 0.169 radians/year
Tangential velocity (vt) ≈ 2.74x10^10 m/year