A planet has a mass of 6.15 × 1023 kg and a radius of 3.25 × 106 m. (a) What is the acceleration due to gravity on this planet? (b) How much would a 79.4-kg person weigh on this planet?
To find the acceleration due to gravity on the planet, we can use the equation:
acceleration due to gravity (g) = G * (mass of the planet) / (radius of the planet)^2
where G is the gravitational constant (approximately 6.674 × 10^(-11) N·m^2/kg^2).
(a) Calculating the acceleration due to gravity:
g = (6.674 × 10^(-11) N·m^2/kg^2) * (6.15 × 10^23 kg) / (3.25 × 10^6 m)^2
Calculating this yields:
g ≈ 1.27 m/s^2
So, the acceleration due to gravity on this planet is approximately 1.27 m/s^2.
(b) To find how much a 79.4-kg person would weigh on this planet, we can use the equation:
weight = mass * acceleration due to gravity
Plugging in the known values:
weight = (79.4 kg) * (1.27 m/s^2)
Calculating this gives:
weight ≈ 100.6 N
Therefore, a 79.4-kg person would weigh approximately 100.6 Newtons on this planet.
To calculate the acceleration due to gravity on a planet, we can use the formula:
\( g = \frac{G \cdot M}{R^2} \)
where:
- \( g \) is the acceleration due to gravity,
- \( G \) is the gravitational constant (\( 6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \)),
- \( M \) is the mass of the planet, and
- \( R \) is the radius of the planet.
(a) To find the acceleration due to gravity on the planet with the given mass and radius, we can substitute these values into the formula:
\( g = \frac{(6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2}) \cdot (6.15 \times 10^{23} \, \text{kg})}{(3.25 \times 10^6 \, \text{m})^2} \)
Calculating this expression will give us the value of the acceleration due to gravity on the planet.
(b) To calculate how much a person weighing 79.4 kg would weigh on this planet, we can use the formula:
\( \text{Weight on planet} = \text{Mass} \times g \)
Substitute the given weight of the person and the calculated acceleration due to gravity in the formula to find the weight on the planet.
F = m g = G M m /R^2
so
g = G M/R^2 =
6.67*10^-11*6.15*10^23/[(3.25^2)*10^12]
= 3.88 * 10^0 = 3.88 m/s^2
about 1/3 of earth g
weight = m g = 79.4 * 3.88 = 308 Newtons