Setting universal gravitation equal to the surface gravitational approximation

Where  will be the surface gravitational acceleration constant

Solving for 

Plugging in values

Setting universal gravitation equal to the gravitational approximation

Where  will be the surface gravitational acceleration constant

Solving for 

Converting  to  and plugging in values

The gravity equation is:

, where  is the universal gravitation constant,  is the mass of the planet, and   is the radius of planet. To find a ratio between the gravity on the planet Zina and our planet, we need to plug in the relevant information:

Remember, we are doing a ratio, so if we reference everything in terms of Earth numbers, the bottom fraction becomes much simpler and cancels out with the top fraction. Also, the gravitational constant is constant, so it cancels out. Therefore, the only important information are the ratios the mass and radius differ from Earth's. 

The radius of the orbit will be the radius of the moon plus the altitude of the orbit.

Converting  to  and plugging in values:

Centripetal force will need to equal universal gravitational force

Solving for velocity

Plugging in values:

To solve this we use the universal gravitation formula

where G is the gravitational constant

and r is the radius of the planet 

plugging everything in we get

This question tests your understanding of gravitational force between two objects, and your ability to apply the formula for gravitational force. 

The formula for gravitational force is as follows: 

In this example, we are asked to explicitly solve for the gravitational force between the two planets in this alternative universe. We are given the values for , , and . To solve for the gravitational force between the planets, we must first calculate the value of , or the distance between the centers of each planet. We are given the values of the diameters of each of the planets, and therefore, if we divide each value by , we have the values of the radii for each planet. To calculate the distance between the centers of the two planets, we must add the values of each planet's radius together, in addition to the distance between the closest surfaces of the planets. Thus, . After completing the arithmetic, you find that . 

Now, you have each of the relevant values to plug into the gravitational force formula. The calculation is shown below:

Therefore, the gravitational force between the two planets is .

This question tests your understanding of gravitational potential energy between two objects, and your ability to apply the formula for gravitational potential energy. 

The formula for gravitational potential energy is as follows: 

In this question, you are asked to solve for the magnitude of the gravitational potential energy between the planet and the star in this alternative universe. We are given the values for , , and . To solve for the gravitational potential energy between the planets, we must first calculate the value of , or the distance between the centers of the planet and the star. We are given the values of the diameters of the planet and the star, and therefore, if we divide each value by , we have the values of the radii for each. To calculate the distance between the centers of each, we must add the values of each radius together, in addition to the distance between the closest surfaces of the two bodies. Thus, . After completing the arithmetic, you find that . 

Now, you have each of the relevant values to plug into the gravitational potential energy formula. The calculation is shown below:

Because we are asked to find the magnitude of the value, we can neglect the negative sign.

Therefore, the magnitude of the gravitational potential energy between the two bodies is .

In orbit, the magnitude of the centripetal force is in magnitude equal to the gravitational force:

Where is the linear velocity and is the distance from the center of the earth.

Solve for :

Plug in values, making sure to convert kilometers to meters to match the units in the answers:

The free body diagram of the system is shown below:

Notice how the friction is pointing downward. This is because it is fighting against the applied upward force. If there were no applied upward force, the force of friction would be pointing upward.

The moment before the block begins to move upward, all vertical forces sum to equal 0. Therefore, we can write:

is the force of friction, and  is the weight of the block

We know that the force of friction can be written as a function of the normal force and coefficient of friction:

Substituting, we get:

We know all the variables on the right, so we can plug in and solve:

You need to have a firm understanding of static friction in order to answer this question correctly.

The main concept covered in this question is that static fricton between two objects has a max and a min, and can have any value between the max and min.

The max static frictional force can be calculated by the equation:

where mu is the coefficient of friction and N is the normal force

In this problem we get:

However, this is not the answer. The problem statement says that the man is pushing with a force of 35N. Since this is less than the max force calculated, the box will not move. Therefore, the force applied from friction is equal to the force applied by the man, 35N. If these forces were not balanced, then there would be a net force in the opposite direction of the man pushing and the box would accelerate due to friction; this is not possible.