find the rotational kinetic energy of the earth about the sun due to its orbital about the sun. The mass of the each M is 6 X 10^24 kg, the orbital radius r is 1.5 X 10^11 m and the rotational period T is 1 year. Hint, treat the earth as a point mass in this problem.
To find the rotational kinetic energy of the Earth about the Sun, we can use the formula:
Rotational Kinetic Energy = (1/2) * Moment of Inertia * Angular Velocity^2
Since we are treating the Earth as a point mass, the moment of inertia can be approximated as:
Moment of Inertia (I) = Mass of the Earth (M) * Distance to the axis of rotation (r)^2
The angular velocity (ω) can be calculated using the formula:
Angular Velocity (ω) = 2π / Time period (T)
Given:
Mass of the Earth (M) = 6 × 10^24 kg
Orbital radius (r) = 1.5 × 10^11 m
Rotational Period (T) = 1 year
Let's substitute these values into the formulas and calculate the rotational kinetic energy step by step:
Step 1: Calculate the moment of inertia (I)
I = M * r^2
= (6 × 10^24 kg) * (1.5 × 10^11 m)^2
Step 2: Calculate the angular velocity (ω)
ω = 2π / T
= 2π / (1 year)
= 2π / (365 × 24 × 60 × 60 s)
Step 3: Calculate the rotational kinetic energy
Rotational Kinetic Energy = (1/2) * I * ω^2
Now, let's substitute the values and evaluate the expression to find the answer.
To find the rotational kinetic energy of the Earth about the Sun, we need to use the formula for rotational kinetic energy:
Rotational kinetic energy = (1/2) * Moment of inertia * Angular velocity^2
Let's break down the problem step by step:
1. Determine the moment of inertia:
In this problem, we are treating the Earth as a point mass, so we can use the formula for the moment of inertia of a point mass:
Moment of inertia (I) = mass * radius^2
Given that the mass of the Earth (M) is 6 x 10^24 kg and the orbital radius (r) is 1.5 x 10^11 m, we can calculate the moment of inertia:
I = M * r^2
2. Find the angular velocity:
The angular velocity (ω) can be calculated using the formula:
Angular velocity (ω) = 2π / Period (T)
Given that the rotational period (T) is 1 year, we need to convert it to seconds first. Since 1 year = 365 days and 1 day = 24 hours and 1 hour = 60 minutes and 1 minute = 60 seconds, we can calculate:
T (in seconds) = 1 year * 365 days * 24 hours * 60 minutes * 60 seconds
3. Calculate the rotational kinetic energy:
Using the moment of inertia (I) and the angular velocity (ω), we can now calculate the rotational kinetic energy using the formula mentioned earlier:
Rotational kinetic energy = (1/2) * I * ω^2
Substitute the values of I and ω into the formula to get the desired answer.
Note: Ensure to use consistent units throughout the calculations to obtain accurate results.