Star Z has 1/5 the radius of Earth and 1000 times the Earth's mass. If a mass weighs 1.0 N on Earth, what does it weight on Star Z?
To find the weight of a mass on Star Z, we need to understand the relationship between weight and mass. The weight of an object is given by the formula:
Weight = Mass × Acceleration due to gravity
On Earth, the acceleration due to gravity is approximately 9.8 m/s². So, if a mass weighs 1.0 N on Earth, we can use the formula to calculate its mass:
Weight on Earth = Mass × Acceleration due to gravity on Earth
1.0 N = Mass × 9.8 m/s²
Solving for Mass, we get:
Mass = 1.0 N / 9.8 m/s²
Now, we can find the weight of the same mass on Star Z. Since the mass on Star Z remains the same, we need to find the acceleration due to gravity on Star Z. The acceleration due to gravity on a celestial body is given by the formula:
Acceleration due to gravity = (Gravitational constant × Mass of the celestial body) / (Radius of the celestial body)²
In this case, Star Z has 1/5 the radius of Earth and 1000 times the Earth's mass. So, let's calculate the acceleration due to gravity on Star Z:
Acceleration due to gravity on Star Z = (Gravitational constant × Mass of Star Z) / (Radius of Star Z)²
Since we are comparing the weight on Star Z to the weight on Earth, we can ignore the gravitational constant as it is the same for both celestial bodies.
Acceleration due to gravity on Star Z = (Mass of Star Z) / (Radius of Star Z)²
= (1000 × Mass of Earth) / [(1/5) × Radius of Earth]²
= (1000 × Mass of Earth) / (1/25) × (Radius of Earth)²
Now, substituting the known values into the equation, we get:
Acceleration due to gravity on Star Z = 1000 × (x / 9.8) / (1/25)
where x is the mass of the Earth.
Simplifying the equation:
Acceleration due to gravity on Star Z = 25,000 × x / 9.8
Now, by substituting the value of Mass on Star Z and the calculated acceleration due to gravity on Star Z back into the weight formula:
Weight on Star Z = Mass on Star Z × Acceleration due to gravity on Star Z
= (x / 9.8) × 25,000 × x / 9.8
Canceling out the factors of 9.8, the weight on Star Z can be simplified to:
Weight on Star Z = 25,000 × x² / 9.8
So, the weight of the same mass on Star Z is 25,000 times greater compared to Earth. Therefore, if a mass weighs 1.0 N on Earth, the weight of the same mass on Star Z would be:
Weight on Star Z = 25,000 × (1.0 N)² / 9.8
= 25,000 / 9.8
≈ 2551.02 N
So, the mass would weigh approximately 2551.02 Newtons on Star Z.