On some planet with a radius of 8,553 km and a mass of 5.55 x 1024 kg the planet is spinning so fast such that anything at the equator will fly off of the planet. What is the planet's period (length of a day on the planet) in hours?
Please help me solve !
To find the planet's period (length of a day on the planet) in hours, we need to use the formula for the period of rotation:
T = 2π√(r^3 / GM)
Where:
T = Period (in seconds)
π = Pi, approximately 3.14159
r = Radius of the planet (in meters)
G = Gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
M = Mass of the planet (in kilograms)
First, let's convert the radius from kilometers to meters:
r = 8,553 km = 8,553,000 meters
Now, let's substitute the given values into the formula and calculate the period:
T = 2π√((8,553,000)^3 / (6.67430 × 10^-11 * 5.55 × 10^24))
Simplifying:
T = 2π√((8,553,000)^3 / (6.67430 × 10^-11 * 5.55 × 10^24))
T = 2π√(6.1439731 x 10^24) / (3.706546x 10^14)
T = 2π√(16604) / (3.706546x 10^14)
T = 2π(128.77802) / (3.706546x 10^14)
T = 807.79973 / (3.706546x 10^14)
T ≈ 2.1780583 x 10^-12
Now, to convert from seconds to hours, we'll divide by 3600 (the number of seconds in an hour):
T (in hours) ≈ (2.1780583 x 10^-12) / 3600
T (in hours) ≈ 6.050161x 10^-16
So, the period (length of a day on the planet) is approximately 6.05 x 10^-16 hours.
To find the planet's period (length of a day), we need to calculate the time it takes for the planet to complete one full rotation.
The formula to calculate the period is: Period = 2π * √(radius / acceleration due to gravity)
Let's break down the steps to calculate the period of the planet:
Step 1: Convert the radius from km to meters.
Given that the radius of the planet is 8,553 km, multiply it by 1000 to convert it to meters:
Radius = 8,553 km * 1000 = 8,553,000 meters
Step 2: Calculate the acceleration due to gravity.
The acceleration due to gravity is determined by the mass of the planet.
Here, the mass of the planet is given as 5.55 x 10^24 kg.
Step 3: Substitute the values into the formula and calculate.
Using the formula: Period = 2π * √(radius / acceleration due to gravity)
Substitute the values:
Period = 2π * √(8,553,000 / (5.55 x 10^24))
Step 4: Simplify and solve for the period.
Calculating the value inside the square root (√) gives us: √(8,553,000 / (5.55 x 10^24)) = 4.61 x 10^-12
Now, substitute this value back into the original equation:
Period = 2π * 4.61 x 10^-12
Solving this equation gives us the period in seconds:
Period = 2π * 4.61 x 10^-12
Period = 2 * 3.14 * 4.61 x 10^-12
Period = 2.90 x 10^-11 seconds
Step 5: Convert the period from seconds to hours.
To convert seconds to hours, divide the period in seconds by 3600 (the number of seconds in an hour):
Period (in seconds) = 2.90 x 10^-11 seconds
Period (in hours) = (2.90 x 10^-11) / 3600
Calculating this gives us the final answer for the period (length of a day on the planet) in hours.