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
C = 2pi*r = 6.28 * 1.62*10^11 = 10.17*10^11m.
Va = 1rev/37yrs * 6.28rad/rev = 0.1697rad/yr. = 9.73deg/yr = Angular velocity.
Vt = 10.17*10^11m/rev * (1/37)rev/yr *
(1/8760)yr/h = 3.138*10^6m/h = 3138km/h
= Tangential velocity.
To determine the quantities for this orbital motion, we can use the equations of circular motion.
1. Angular acceleration (α):
Angular acceleration (α) can be found using the formula:
α = (2π) / T,
where T is the period of revolution in seconds.
In this case, the period of revolution is given as 37.0 years. To convert this to seconds, we multiply by the number of seconds in a year:
T = 37.0 years x 365 days/year x 24 hours/day x 60 minutes/hour x 60 seconds/minute.
Substituting the value of T into the formula, we get:
α = (2π) / (37.0 years x 365 days/year x 24 hours/day x 60 minutes/hour x 60 seconds/minute).
2. Tangential acceleration (at):
Tangential acceleration (at) can be calculated using the formula:
at = r * α,
where r is the radius of the circular orbit.
In this case, the radius of the circular orbit is given as 1.62x10^11 m. Substituting the value of α into the formula, we get:
at = (1.62x10^11 m) * α.
3. Radial acceleration (ar):
Radial acceleration (ar) is the centripetal acceleration and can be calculated using the formula:
ar = (v^2) / r,
where v is the velocity and r is the radius.
4. Angular velocity (ω):
Angular velocity (ω) can be found using the formula:
ω = (2π) / T.
5. Tangential velocity (vt):
Tangential velocity (vt) can be calculated using the formula:
vt = ω * r.
Substituting the values given for radius and angular velocity into the formula, we get:
vt = (2π / T) * (1.62x10^11 m).