Physics, 19.01.2020 05:31 weeman67161

Can someone check if i did this right? question: derive a formula for the mass of a planet in terms of its radius, r, the acceleration due to gravity at its surface, gp, and the gravitational constant, g.

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The expression for the mass of the planet in terms of the radius of the planet, acceleration due to gravity and gravitational constant is $\boxed{{M_P} = \dfrac{{{(g_p)}{r^2}}}{G}}$ .

Further explanation:

Here, we have to derive the expression for the mass of the planet in terms of the radius of the planet, acceleration due to gravity and gravitational constant.

From the Newton’s law of the gravitation, gravitational force exerted by the planet on the object at the surface of the planet is directly proportional to the product of the mass of the planet and the object and inversely proportional to the square of the distance between them.

It can be written mathematically as,

$\boxed{F = \frac{{G{M_e}{m_o}}}{{{r^2}}}}$

Here, is the gravitational constant and its value is $6.674 \times {10^{ - 11}}\text{ m}^3/\text{kg}\cdot\text{s}^2$ , ${M_e}$ is the mass of the planet, ${m_o}$ is the mass of object, is the radius of the planet.

So, the gravitational force exerted by the planet on the object of unit mass at the surface of the planet can be calculated as,

${F_1}=\dfrac{{G{M_P}}}{{{r^2}}}$

As we know, the acceleration due to gravity is equal to the gravitational force or weight of the unit mass of object.

So, the acceleration due to gravity of the planet can be expressed as,

${g_p} = {F_1}$

Substitute the value of the ${F_1}$ as $\dfrac{{G{M_P}}}{{{r^2}}}$ in the above equation.

${g_p}=\dfrac{{G{M_P}}}{{{r^2}}}$

Simplify the above equation for mass of the planet,

$\boxed{{M_P} = \frac{{{g_p}{r^2}}}{G}}$

Thus, the expression for the mass of the planet in terms of the radius of the planet, acceleration due to gravity and gravitational constant is $\boxed{{M_P} = \dfrac{{{(g_p)}{r^2}}}{G}}$ .

Learn more:

1. Acceleration of object against friction:

2. Change in the behavior of object due to change in gravity:

Answer detail:

Grade: Senior School

Subject: Physics

Chapter: Gravitation

Keywords:

Derive a formula, mass of planet, radius, gravity, gravitational constant , gravitation, Earth, 6.67x10^-11, acceleration due to gravity, force of attraction, object.

Answer from: Quest

its c its is a mixture of gases

Answer from: Quest

force is based upon mass and acceleration.the statement is not always true.

it can be explained from newton's second law of motion.

as per newton's second law of motion the rate of change of momentum is directly proportional to the applied force and takes place along the direction of force.

mathematically it can be written as -

$f\alpha \frac{dp}{dt}$

⇒ $f= k \frac{dp}{dt}$

here k is the proportionality constant whose value is 1.

$f=\frac{dp}{dt}$

$f= \frac{d[mv]}{dt}$

$f= m\frac{dv}{dt} +v\frac{dm}{dt}$

$f=ma+v\frac{dm}{dt}$ [a is the acceleration=dv/dt]

hence we see that force not only depends on mass and acceleration but also on velocity and rate of change of mass. force f will depend on mass and acceleration only when mass is constant.

hence the statement that force is based upon mass and acceleration is false.

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Can someone check if i did this right? question: derive a formula for the mass of a planet in term...