Suppose a new extrasolar planet is discovered. Its mass is double the mass of the Earth, but it has the same density and spherical shape as the Earth. How would the weight of an object at the new planet's surface differ from its weight on Earth? (Let Wnew be the weight of the object at the new planet's surface and WE be the weight of the object at the surface of the Earth. Use any variable or symbol stated above as necessary.)
g(n) = G*Mn/Rn^2 & g(e)= G*Me/Re^2
Mn=d*(4/3)*pi*Rn^3 (Mn->new planet's mass)
Me=d*(4/3)*pi*Re^3 (Me-> Earth's mass)
So, g(n)/g(e) = Rn/Re .....(1)
Now, Mn/Me = 2 (given)
So Rn^3/Re^3 = 2
or Rn/Re = 2^1/3 ......(2)
From (1) & (2)
g(n)/g(e)= Rn/Re = 2^1/3
So, if the mass of the object is m:
Wn/We = m*g(n)/(m*g(e))
= 2^1/3
To determine how the weight of an object at the new planet's surface differs from its weight on Earth, we can use the following equation:
Wnew = m * gnew
where Wnew is the weight of the object at the new planet's surface, m is the mass of the object, and gnew is the acceleration due to gravity at the new planet's surface.
Similarly, the weight of the object on Earth can be calculated using the equation:
WE = m * gE
where WE is the weight of the object on Earth, gE is the acceleration due to gravity on Earth (approximately 9.8 m/s^2).
Since the new planet has double the mass of the Earth but the same density, we can conclude that the new planet's radius must be approximately ∛2 times that of Earth, assuming the objects have the same density.
Now, the equation for the acceleration due to gravity at the surface of a planet can be given as:
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 density and shape of the new planet are the same as Earth, we can assume that the radius of the new planet is ∛2 times that of Earth.
Substituting these values into the equation, we have:
gnew = (G * (2M)) / ((∛2 * R)^2)
= (G * 2M) / (2 * R^2)
= (G * M) / R^2
= gE
Therefore, the acceleration due to gravity at the new planet's surface is the same as that of Earth.
This means that the weight of an object at the new planet's surface would be the same as its weight on Earth. In equation form:
Wnew = m * gnew = m * gE = WE
So, the weight of an object at the new planet's surface would be the same as its weight on Earth.
To determine how the weight of an object at the new planet's surface differs from its weight on Earth, we need to consider Newton's Law of Universal Gravitation and the concept of gravitational acceleration.
The formula for the weight of an object can be given as:
Weight = mass × gravitational acceleration
Now, the mass of the object remains constant regardless of the planet, so we only need to compare the gravitational accelerations on Earth and the new planet.
The gravitational acceleration on a planet is given by the formula:
gravitational acceleration = (gravitational constant × mass of the planet) / (radius of the planet)^2
Since the density and shape of the new planet are the same as Earth, we can assume that the radii of both planets are the same. Let's call this radius "R".
Now, let's calculate the gravitational acceleration on Earth and the new planet:
For Earth:
gravitational acceleration on Earth = G × mass of Earth / R^2
Let's call this value "gE".
For the new planet:
gravitational acceleration on the new planet = G × (mass of the new planet) / R^2
Let's call this value "gnew".
Given that the mass of the new planet is double the mass of Earth, we can denote it as "2M".
gravitational acceleration on the new planet = G × 2M / R^2
Comparing the gravitational accelerations on Earth and the new planet:
gnew = G × 2M / R^2
gE = G × M / R^2
Now, we can determine the weight of the object at the new planet's surface (Wnew) and compare it to the weight of the object on Earth (WE).
Considering that the weight is calculated based on mass and gravitational acceleration, we can express the weight of objects as:
Wnew = mass × gnew
WE = mass × gE
Since the mass of the object remains constant, we can cancel it in the equations:
Wnew = mass × gnew = mass × G × 2M / R^2 = 2MG / R^2
WE = mass × gE = mass × G × M / R^2 = MGM / R^2
Therefore, the weight of the object at the new planet's surface (Wnew) would be 2 times the weight of the object on Earth (WE).