26. If both the mass and radius of earth decrease by 1%, the value of ...
Answer:Introduction
In this question, we are given that both the mass and radius of the Earth decrease by 1%. We need to determine the effects of this change on the following variables:
(I) acceleration due to gravity
(II) escape velocity from Earth's surface
(III) gravitational potential energy of a body on Earth's surface
Effects on acceleration due to gravity
The acceleration due to gravity is given by the equation:
g = G * (M / R^2)
where:
g is the acceleration due to gravity
G is the gravitational constant
M is the mass of the Earth
R is the radius of the Earth
Since both the mass and radius of the Earth decrease by 1%, the equation becomes:
g' = G * ((0.99M) / (0.99R)^2)
Simplifying the equation, we find that:
g' = (0.99^2) * (G * (M / R^2))
Therefore, the acceleration due to gravity decreases by (1 - 0.99^2) = 0.0199, or approximately 1.99%. Thus,
(II) acceleration due to gravity would decrease by 1%.Effects on escape velocity
The escape velocity from Earth's surface is given by the equation:
v = √((2 * G * M) / R)
where:
v is the escape velocity
G is the gravitational constant
M is the mass of the Earth
R is the radius of the Earth
Since both the mass and radius of the Earth decrease by 1%, the equation becomes:
v' = √((2 * G * (0.99M)) / (0.99R))
Simplifying the equation, we find that:
v' = √(((0.99^2) * (2 * G * M)) / (0.99^2 * R))
Therefore, the escape velocity decreases by a factor of 0.99, or approximately 1%. Thus,
(III) escape velocity from Earth's surface would decrease by 1%.Effects on gravitational potential energy
The gravitational potential energy of a body on Earth's surface is given by the equation:
PE = - (G * M * m) / R
where:
PE is the gravitational potential energy
G is the gravitational constant
M is the mass of the Earth
m is the mass of the body
R is the radius of the Earth
Since both the mass and radius of the Earth decrease by 1%, the equation becomes:
PE' = - (G * (0.99M) * m) / (0.99R)
Simplifying the equation, we find that:
PE' = (0.99^2) * (- (G * M * m) / R)
Therefore, the gravitational potential energy remains unchanged. Thus,
(IV) the gravitational potential energy of a body on Earth's surface will remain unchanged.Conclusion
Based on the calculations above, we can conclude that the correct answer is option
(II) and (IV). The acceleration due to gravity would decrease by 1% and the gravitational potential energy of a body