IMAGINE- a building 6400km. high. On the ground floor, a person weighs 175lbs when he steps on a spring scale how much would the man weigh on the same scale if he were standing at the top floor? HINT: Note that 6400km is also the radius of the earth so that the top floor is 2Re from the earths center. Think about what this means for Fgrav before you start plugging in numbers or making conversions
I KNOW THIS MUCH----- confuesed what to plug in where. and what not to plug in. The weight is equal ‘mg”
On the Earth surface acceleration due to gravity is
mg=GmM/R²,
g=G•M•/R²,
where
the gravitational constant G =6.67•10^-11 N•m²/kg²,
Earth’s mass is M = 5.97•10^24 kg,
Earth’s radius is R = 6.378•10^6 m.
g=9.8 m/s²
Let's find “g” at the height “h” . Distance between the mas “m” (the person) and the center of the Earth is 2R =>
m•g1= G•m•M/(2R)²
g1= G•M/4R²=g/4
In the "high" building mg1= mg/4
You are making this much too complicated.
This is right:
mg=GmM/R²
now what if to went to radius = 2R
then
mg = GmM/(2R)^2
= GmM/4R^2
or
new weight = 1/4 of original weight
That is what their HINT means.
To solve this problem, let's start by understanding the concept of gravitational force. Gravitational force is directly proportional to the mass of an object and inversely proportional to the square of the distance between the object and the center of the Earth.
In this scenario, the person's weight is measured using a spring scale, which gives a reading based on the force exerted downwards due to gravity. So, let's assume the person's mass remains constant.
Given that the building is 6400 km high, which is the same as the Earth's radius, we can calculate the distance between the person on the ground floor and the person on the top floor.
On the ground floor, the distance from the center of the Earth is Re (Earth's radius). On the top floor, the distance is twice the Earth's radius, 2Re. Let's denote the weight on the top floor as W2.
The gravitational force (Fgrav) on the top floor is proportional to the mass (m) of the person and the inverse square of the distance (d) between the person and the center of the Earth:
Fgrav ∝ m/d^2
Now, we can set up a proportion using the concept of gravitational force:
W1/W2 = (m/d1^2) / (m/d2^2)
In this case, W1 represents the weight on the ground floor, m represents the mass of the person, d1 represents the distance from the person on the ground floor to the center of the Earth (Re), and d2 represents the distance from the person on the top floor to the center of the Earth (2Re).
Simplifying the proportion:
W1/W2 = (1/d1^2) / (1/d2^2)
W1/W2 = d2^2 / d1^2
Now, let's plug in the values:
W1 = 175 lbs
d2 = 2Re = 2 * 6400 km (note: I'll convert this to the appropriate unit)
d1 = Re = 6400 km (note: I'll convert this to the appropriate unit)
First, we need to convert the values of Re, d1, and d2 into a consistent unit of measurement. Let's use meters instead of kilometers:
Re = 6400 km * 1000 m/km = 6,400,000 m
d1 = Re = 6,400,000 m
d2 = 2Re = 2 * 6,400,000 m = 12,800,000 m
Plugging in the values into the proportion:
175/W2 = (12,800,000^2) / (6,400,000^2)
Now, we can solve for W2:
W2 = 175 lbs * [(6,400,000 m)^2] / [(12,800,000 m)^2]
Calculating further:
W2 = 175 lbs * (40^2) / (80^2)
W2 = 175 lbs * 0.5
W2 = 87.5 lbs
Therefore, on the same scale, the person would weigh 87.5 pounds when standing at the top floor.
Note: It's important to remember that this calculation assumes a constant mass for the person and ideal conditions as a simplification. In practice, weight could vary slightly due to other factors like altitude, air resistance, and local gravity variations.