A planet orbits a star, in a year of length 4.46 x 107 s, in a nearly circular orbit of radius 3.17 x 1011 m. With respect to the star, determine (a) the angular speed of the planet, (b) the tangential speed of the planet, and (c) the magnitude and direction of the planet's centripetal acceleration.
To determine the angular speed, tangential speed, and centripetal acceleration of the planet, we can use the following formulas:
(a) Angular Speed:
The angular speed (ω) can be calculated using the formula:
ω = 2π / T
where T is the period of the orbit, given as 4.46 x 10^7 s.
Substituting the given values, we get:
ω = 2π / (4.46 x 10^7 s)
(b) Tangential Speed:
The tangential speed (v) of the planet can be calculated using the formula:
v = 2πr / T
where r is the radius of the orbit, given as 3.17 x 10^11 m.
Substituting the given values, we get:
v = 2π(3.17 x 10^11 m) / (4.46 x 10^7 s)
(c) Centripetal Acceleration:
The centripetal acceleration (ac) of the planet can be calculated using the formula:
ac = v^2 / r
Substituting the given values, we get:
ac = (2π(3.17 x 10^11 m) / (4.46 x 10^7 s))^2 / (3.17 x 10^11 m)
Now let's calculate the values step-by-step:
(a) Angular Speed:
ω = 2π / (4.46 x 10^7 s)
= 1.411 x 10^-7 rad/s
(b) Tangential Speed:
v = 2π(3.17 x 10^11 m) / (4.46 x 10^7 s)
= 1.419 x 10^4 m/s
(c) Centripetal Acceleration:
ac = (2π(3.17 x 10^11 m) / (4.46 x 10^7 s))^2 / (3.17 x 10^11 m)
= 4.745 x 10^-2 m/s^2
Therefore, the results are:
(a) The angular speed of the planet is approximately 1.411 x 10^-7 rad/s.
(b) The tangential speed of the planet is approximately 1.419 x 10^4 m/s.
(c) The magnitude of the planet's centripetal acceleration is approximately 4.745 x 10^-2 m/s^2, and it acts towards the center of the orbit.
To determine the answers to the given questions, we need to use the formulas related to rotational motion and circular motion.
(a) The angular speed of the planet can be found using the formula:
angular speed (ω) = 2π / T,
where T is the time period (year) in seconds.
Plugging in the values, we have:
ω = 2π / (4.46 x 10^7 s)
Calculate this value to find the angular speed of the planet.
(b) The tangential speed of the planet can be calculated using the formula:
tangential speed (v) = ω x r,
where ω is the angular speed and r is the radius of the orbit.
Plug in the values of ω and r to find the tangential speed of the planet.
(c) The centripetal acceleration can be determined using the formula:
centripetal acceleration (a) = v^2 / r,
where v is the tangential speed and r is the radius of the orbit.
Plug in the value of v and r to calculate the magnitude of the centripetal acceleration.
To find the direction of the planet's centripetal acceleration, we need to consider that it is directed towards the center of the circular orbit.
Calculate the magnitude and direction of the planet's centripetal acceleration using the above formula.