The free-fall acceleration at the surface of planet 1 is 18 m/s^2. The radius and the mass of planet 2 are twice those of planet 1. What is the free-fall acceleration on planet 2?
Well, if the radius and the mass of planet 1 are twice those of planet 2, then it must mean that planet 2 has a serious case of shrinking and dieting. It's like a planet makeover!
But let's get to the serious stuff. The free-fall acceleration is given by the equation: g = (G * M) / R^2, where G is the gravitational constant, M is the mass of the planet, and R is the radius of the planet.
Since the radius and mass of planet 2 are twice those of planet 1, we can say that R2 = 2 * R1 and M2 = 2 * M1.
Now, let's substitute the values into the equation for planet 1: g1 = (G * M1) / R1^2.
For planet 2, we have: g2 = (G * M2) / R2^2.
Substituting the values we know for planet 2: g2 = (G * 2 * M1) / (2 * R1)^2.
Simplifying that equation: g2 = (G * 2 * M1) / 4 * R1^2.
And since we know that g1 = 18 m/s^2, we can substitute that value: 18 = (G * M1) / R1^2.
Let's rearrange the equation to solve for G: G = 18 * R1^2 / M1.
Now, we can plug this value of G back into the equation for planet 2: g2 = (18 * R1^2 / M1) * 2 * M1 / (4 * R1^2).
Simplifying the equation for planet 2: g2 = (9 * R1^2) / (2 * R1^2).
And finally, canceling out the R1^2 terms, we get: g2 = 9 / 2.
So, the free-fall acceleration on planet 2 is 9 / 2 m/s^2. It's as simple as pie... or should I say, planetary pie?
To find the free-fall acceleration on planet 2, we can use the formula for gravitational acceleration:
g = (G * M) / R^2
where:
g is the gravitational acceleration,
G is the gravitational constant,
M is the mass of the planet, and
R is the radius of the planet.
Given that the radius and mass of planet 2 are twice those of planet 1, we can write:
R2 = 2 * R1
M2 = 2 * M1
Let's substitute these values into the formula for gravitational acceleration:
g2 = (G * M2) / R2^2
Since R2 = 2 * R1 and M2 = 2 * M1, we can substitute these equations into the above equation:
g2 = (G * (2 * M1)) / ((2 * R1)^2)
Now, let's simplify the equation:
g2 = (G * 2 * M1) / (4 * R1^2)
g2 = (G * M1) / (2 * R1^2)
Given that the free-fall acceleration on planet 1 is 18 m/s^2, we have:
g1 = 18 m/s^2
Now, let's substitute the values of g1, G, M1, and R1 into the equation above:
18 = (G * M1) / (2 * R1^2)
Finally, let's solve for g2:
g2 = (18 * 2 * R1^2) / (G * M1)
Since G, M1, and R1 are constants, we can simplify the equation further:
g2 = 36 * R1^2 / (G * M1)
Therefore, the free-fall acceleration on planet 2 is 36 times the free-fall acceleration on planet 1.
To find the free-fall acceleration on planet 2, we can use the equation for gravitational acceleration:
a = GM/r^2
where:
a is the free-fall acceleration,
G is the universal gravitational constant (approximately 6.67 x 10^-11 m^3/kg/s^2),
M is the mass of the planet, and
r is the radius of the planet.
Given that the mass and radius of planet 2 are twice those of planet 1, we can write the following relationships:
M2 = 2M1
r2 = 2r1
Assuming the free-fall acceleration on planet 1 as 18 m/s^2, we can substitute the values into the equation and solve for the free-fall acceleration on planet 2:
a2 = G(M2/r2^2)
= G((2M1)/(2r1)^2)
= G(M1/r1^2)
= a1
Therefore, the free-fall acceleration on planet 2 is also 18 m/s^2.
g= GM/R^2
For Planet 1, Let it be g1=GM1/R1^2
And for planet 2, g2=GM2/R2^2[M2=2M1 and R2= 2 R1]
So g2=GM1/2R1^2
g2=g1/2=9m/s^2.
Hope this helped!!!