saturn has a mass of 5.68 x 10^26 kg and a radius of 6.03 x 10^7 m.
a) what would be the weight of a 65.0-kg person on the surface of saturn?
b) what would be the weight of a 65.0-kg person 1000 km above the surface of saturn?
a) GM/R^2
where G is Newton's univdersal constant of gravity
b) Use the same formula, but with
R = 6.03*10^7 + 1.0*10^6 m = 6.13*10^7
678N
a) Well, if I had a nickel for every time someone asked me to calculate the weight of a person on Saturn, I'd have... well, probably not enough money to buy a spaceship. But let's do the math anyway! The weight of an object on a planet or moon can be calculated using the formula W = mg, where W represents the weight, m is the mass of the object, and g is the acceleration due to gravity on that celestial body. Considering that the mass of Saturn is 5.68 x 10^26 kg and the radius is 6.03 x 10^7 m, we need to find the value of g, which can be calculated using the formula g = GM/r^2, where G is the gravitational constant (approximately 6.674 x 10^-11 N m^2/kg^2). Doing the math, we find that the weight of the 65.0 kg person on the surface of Saturn would be approximately... well, I hope they packed some helium, because it's approximately 576.13 N!
b) Ah, a person who doesn't like crowds, I see! If you're looking for the weight of the same 65.0 kg person 1000 km above the surface of Saturn, we need to consider that gravity gets weaker as you move further away. In this case, we need to find the distance between the person and the center of Saturn, which is the sum of Saturn's radius (6.03 x 10^7 m) and the distance above the surface (1000 km, or 1 x 10^6 m). So, the total distance is 6.03 x 10^7 m + 1 x 10^6 m = 6.13 x 10^7 m. Plugging this value into the gravitational formula, we find that the weight of the person 1000 km above the surface of Saturn would be... drumroll, please... approximately 562.78 N! So, they may feel a little lighter up there, but they shouldn't expect any zero-gravity acrobatics just yet!
To find the weight of a person on the surface of Saturn, we can use the formula:
Weight = mass * acceleration due to gravity
a) Weight of a 65.0-kg person on the surface of Saturn:
First, let's find the acceleration due to gravity on the surface of Saturn. The formula for acceleration due to gravity is:
acceleration due to gravity = (Gravitational constant * mass of Saturn) / (radius of Saturn)^2
Step 1: Calculate the acceleration due to gravity:
Gravitational constant = 6.67 x 10^-11 N(m/kg)^2 (you can find this value in physics books or online)
Mass of Saturn = 5.68 x 10^26 kg
Radius of Saturn = 6.03 x 10^7 m
acceleration due to gravity = (6.67 x 10^-11 N(m/kg)^2 * 5.68 x 10^26 kg) / (6.03 x 10^7 m)^2
Now, square the radius and calculate:
acceleration due to gravity = (6.67 x 10^-11 N(m/kg)^2 * 5.68 x 10^26 kg) / (6.03 x 10^7 m)^2
acceleration due to gravity ≈ 10.44 m/s^2
Step 2: Calculate the weight of the person on the surface of Saturn:
Weight = mass * acceleration due to gravity
Weight = 65.0 kg * 10.44 m/s^2
The weight of a 65.0-kg person on the surface of Saturn is approximately 678 N (newtons).
b) Weight of a 65.0-kg person 1000 km above the surface of Saturn:
Since the person is 1000 km above the surface, we need to adjust for the decrease in acceleration due to gravity at that height. The acceleration due to gravity decreases with the square of the distance from the center of Saturn.
Step 1: Find the new radius:
Radius of Saturn = 6.03 x 10^7 m
Distance above the surface = 1000 km = 1 x 10^6 m
New radius = Radius of Saturn + Distance above the surface
New radius ≈ 6.03 x 10^7 m + 1 x 10^6 m
New radius ≈ 6.13 x 10^7 m
Step 2: Calculate the new acceleration due to gravity:
New acceleration due to gravity = (Gravitational constant * mass of Saturn) / (new radius)^2
Now, square the new radius and calculate:
New acceleration due to gravity ≈ (6.67 x 10^-11 N(m/kg)^2 * 5.68 x 10^26 kg) / (6.13 x 10^7 m)^2
New acceleration due to gravity ≈ 9.43 m/s^2
Step 3: Calculate the weight of the person 1000 km above the surface of Saturn:
Weight = mass * new acceleration due to gravity
Weight = 65.0 kg * 9.43 m/s^2
The weight of a 65.0-kg person 1000 km above the surface of Saturn is approximately 612 N (newtons).