Understanding Gravity and Weight

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A satellite orbits above the Earth. What is the satellite's acceleration due to gravity?

$m_E=5.97*10^{24}kg$

$G=6.67*10^{-11}\frac{m^3}{kg*s}$

$2.7*10^{-3}\frac{m}{s^2}$

$9.81\frac{m}{s^2}$

$6.87\frac{m}{s^2}$

$6.24*10^8\frac{m}{s^2}$

$1.32*10^{-5}\frac{m}{s^2}$

Explanation

To solve this problem, use the law of universal gravitation.

$F_G=G\frac{m_1m_2}{r^2}$

Remember that is the distance between the centers of the two objects. That means it will be equal to the radius of the earth PLUS the orbiting distance.

Use the given values for the masses of the objects and distance to solve for the force of gravity.

$F_G=G\frac{m_sm_E}{(r_E+h)^2}$

$F_G=(6.67*10^{-11}\frac{m^3}{kg*s})*\frac{(2500kg)(5.97*10^{24}kg)}{(6.37*10^5m+3.8*10^8m)^2}$

$F_G=(6.67*10^{-11}\frac{m^3}{kg*s^2})*\frac{1.49*10^{28}kg^2}{1.45*10^{17}m^2}$

$F_G=(6.67*10^{-11}\frac{m^3}{kg*s^2})*(1.03*10^{11}\frac{kg^2}{m^2})$

Now that we know the force, we can find the acceleretion. Remember that weight is equal to the mass times acceleration due to gravity.

Set our two forces equal and solve for the acceleration.

$2.7*10^{-3}\frac{m}{s^2}=g$

An astronaut weighs on Earth. On a distant moon, she weighs . What is the acceleration due to gravity on this moon?

$g=-9.8\frac{m}{s^2}$

$8.5\frac{m}{s^2}$

$13.3\frac{m}{s^2}$

$15.3\frac{m}{s^2}$

$0.12\frac{m}{s^2}$

$2.04\frac{m}{s^2}$

Explanation

First we need to find the mass of the astronaut using Newton's second law.

We know the total weight of the astronaut and the acceleration due to gravity on Earth, allowing us to solve for her mass.

$150N=m*9.8\frac{m}{s^2}$

$\frac{150N}{9.8\frac{m}{s^2}}=m$

Now that we know her mass, we can look at her weight on the distant moon. We know her weight and mass, allowing us to solve for the acceleration due to gravity in this new environment.

$\frac{130N}{15.3kg}=a$

$8.5\frac{m}{s^2}=a$

A ball that weighs on Earth weighs on a recently discovered planet. What is the force of gravity on this new planet?

Give your answer with the correct number of significant figures.

$50\frac{m}{s^2}$

$47.5\frac{m}{s^2}$

$48\frac{m}{s^2}$

$50.0\frac{m}{s^2}$

$48.0\frac{m}{s^2}$

Explanation

Weight is defined as the force of gravity on an object. We can use Newton's second law to write an equation for weight.

If the ball weighs on Earth, then its mass can be found using this equation and the acceleration of gravity on Earth.

$4N = m(9.8\frac{m}{s^2})$

$m = \frac{4N}{(9.8\frac{m}{s^2})} = 0.40816 kg$

Use this mass and the given weight on the new planet to find the acceleration of gravity on this new planet. Though our initial values (and thus our final values) only allow one significant figure, we will not round until the end of all calculations. This ensures that we preserve accuracy before adjusting for precision.

$\frac{19N}{0.40816kg} = 46.55\frac{m}{s^2}$

Adjust this value to one significant figure by rounding up. The zero in the tens place is before the decimal, and is not considered significant.

$46.55\frac{m}{s^2}\approx 50\frac{m}{s^2}$

A ball falls off a cliff. What is the force of gravity on the ball?

$g=-9.8\frac{m}{s^2}$

We need to know the height of the cliff in order to solve

We need to know the time the ball is in the air in order to solve

Explanation

Newton's second law states:

In this case the acceleration will be the constant acceleration due to gravity on Earth.

Use the acceleration of gravity and the mass of the ball to solve for the force on the ball.

$F=(0.05kg)(-9.8\frac{m}{s^2})$

The answer is negative because the force is directed downward. Since gravity is always acting downward, a force due to gravity will always be negative.

An astronaut lands on a new planet and discovers her weight on this planet is half of her weight on Earth. What is the acceleration due to gravity on this planet in terms of the acceleration due to gravity on Earth ( $\small g$ )?

$\frac{1}{2}g$

$\sqrt{g}$

We need to know her mass in order to solve

Explanation

Weight is a very specific force, equal to the mass times gravity:

On this new planet her weight is half of what it was on Earth. We can write this mathematically, using the weight equation.

$F_{W2}=\frac{1}{2}F_{W1}$

$mg_{new}=\frac{1}{2}mg_{Earth}$

Mass cancels out from each side, leaving a relationship between the gravitational accelerations.

$g_{new}= \frac{1}{2}g_{Earth}$

That means the acceleration due to gravity on this new planet is half of what it was on earth.

An object is placed in the direct center of the Earth. What would be the perceived weight of the object?

The object would have no weight

The object would have infinite weight upward

The object would have infinite weight downward

We must know the mass of the object to draw a conclusion

The weight of the object would be equal to its weight at the Earth's surface

Explanation

We must use Newton's law of universal gravitation to solve this question.

$F_G=G\frac{m_1m_2}{r^2}$

There are three variables that really change the force of gravity: the mass of each object and the distance between the bodies.

The important thing to recognize here, though, is that when an object is in the center of the earth, the mass of the earth is distributed symmetrically all around it. It's like being in the center of a giant bubble. Because the mass is symmetrically distributed, the mass that is trying to pull the object in each direction is equal. Essentially, the mass pulling upward cancels out the mass pulling downward, and the mass pulling right cancels out the mass pulling left.

This happens for the entirety of the circle, leaving you with a net force of zero acting upon the object.

The acceleration of gravity on the moon is significantly less than the acceleration of gravity on earth. What will happen to an astronaut's weight and mass on the moon, compared to her weight and mass on Earth?

Her weight will decrease and her mass will remain the same

Her weight and mass will decrease

Her weight and mass will remain the same

Her weight will increase, but her mass will decrease

Her weight will remain the same and her mass will decrease

Explanation

Mass is a measure of how much matter is in an object, while weight is a measurement of the effective force of gravity on the object.

The amount of matter in the astronaut does not change; therefore, she has the same mass on the moon as she has on Earth.

Her weight, however, will change due to the change in gravity. The force of gravity will be the product of the acceleration and the astronaut's mass.

If the gravity is less on the moon than on Earth, then the force of gravity on the astronaut will also be less on the moon; thus, she will weigh less.

A satellite orbits above the Earth. What is the gravitational acceleration on the Earth caused by the satellite?

$m_E=5.97*10^{24}kg$

$G=6.67*10^{-11}\frac{m^3}{kg*s}$

$1.15*10^{-24}\frac{m}{s^2}$

$6.87\frac{m}{s^2}$

$2.7*10^{-3}\frac{m}{s^2}$

$0\frac{m}{s^2}$

$9.81\frac{m}{s^2}$

Explanation

To solve this problem, use the law of universal gravitation.

$F_G=G\frac{m_1m_2}{r^2}$

Remember that is the distance between the centers of the two objects. That means it will be equal to the radius of the earth PLUS the orbiting distance.

Use the given values for the masses of the objects and distance to solve for the force of gravity.

$F_G=G\frac{m_sm_E}{(r_E+h)^2}$

$F_G=(6.67*10^{-11}\frac{m^3}{kg*s})*\frac{(2500kg)(5.97*10^{24}kg)}{(6.37*10^5m+3.8*10^8m)^2}$

$F_G=(6.67*10^{-11}\frac{m^3}{kg*s^2})*\frac{1.49*10^{28}kg^2}{1.45*10^{17}m^2}$

$F_G=(6.67*10^{-11}\frac{m^3}{kg*s^2})*(1.03*10^{11}\frac{kg^2}{m^2})$

Now that we know the force, we can find the acceleretion. Remember that weight is equal to the mass times acceleration due to gravity.

Set our two forces equal and solve for the acceleration.

$6.87N=5.97*10^{24}kg*g$

$\frac{6.87N}{5.97*10^{24}kg}=g$

$1.15*10^{-24}\frac{m}{s^2}=g$

A woman stands on the edge of a cliff and drops two rocks, one of mass and one of , from the same height. Which one experiences the greater acceleration?

They experience the same acceleration

The rock with mass

We need to know the height of the cliff in order to solve

We need to know the density of the rocks in order to solve

Explanation

Even though the rocks have different masses, the acceleration on both will be , the acceleration due to gravity. We can look at Newton's second law to see the force experienced by the rocks:

When objects are in free-fall, the acceleration will be equal to the acceleration from gravity, regardless of the mass of the object.

A woman stands on the edge of a cliff and drops two rocks, one of mass and one of , from the same height. Which one experiences the greater force?

The rock with mass

They both experience the same force

We need to know the height of the cliff to determine the answer

We need to know the volume of the rocks to determine the answer

Explanation

The formula for force is given by Newton's second law:

Both rocks will experience the same acceleration, , or the acceleration due to gravity.

Use the mass of each rock in this equation to find which rock experiences a greater force.

$F_{g,m}=(m)g$

$F_{g,3m}=(3m)g=3(mg)$

We can see that the force on the rock with mass of is equal to three times for the force on the rock with mass of . The heavier rock experiences the greater force.

$F_{g,3m}=3(F_{g,m})$

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