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.

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.

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.

Consider a person standing on a scale in an elevator that is not moving.  The person is exerting a downward force onto the scale equal to their weight force.  The scale exerts a force upward that is equal to the downward force of gravity.  This upward force is the reading on the scale.

Now consider a person standing on a scale in an elevator that is moving at a constant velocity.  The person is exerting a downward force onto the scale equal to the force of gravity.  The scale exerts a force upward.  Because the elevator is moving at a constant velocity, these forces are balanced, and the scale would read the same as if you were not moving.

Now consider a person standing in an elevator that is accelerating.  There are still two forces, the downward force of gravity, and the upward normal force.  Since the elevator is accelerating, these forces are unbalanced in the direction of the motion.  If the elevator is accelerating down, gravity is larger than the normal force.  If the elevator is accelerating up, the normal force is greater than gravity.

Since the scale reads the magnitude of the normal force, the time when the normal force is the greatest is when the elevator is accelerating in an upward direction.

If the velocity on an object is constant, that means it has no acceleration. If it has no acceleration, that means that the net force on the object is equal to zero. We can see this conclusion by using Newton's second law.

Another way to think of  is the sum of all the forces. Since the only two forces acting upon the ball are gravity and Jerry's lifting force, we can see: .

Since the net force is zero, the magnitude of Jerry's force must equal the magnitude of the force of gravity, but in the opposite direction.

This means that once we find , then Jerry's lifting force will be the same magnitude but in the opposite direction. Use Newton's second law to find the force of gravity.

This means that Jerry's lifting force will be .

Weight is a particular force that is equal to mass times acceleration due to gravity. Start with Newton's second law,  .

Plug in the given values to solve.

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.

To determine the net force on a system, it is important to consider all the forces between each object in the horizontal direction.  The system is considered to be all the blocks together as they will all move with the same acceleration.

According to Newton’s 3rd Law  and  are equal in magnitude and opposite in direction so they will cancel out.  The same goes for  and .

Therefore our equation is reduce to 

From Newton’s 2nd law we know that the net force on a system is equal to the mass of the system times the acceleration of the system

Therefore

This means that the acceleration on any one block is

So when solving for the net force on any one block, the net force on the single block will be equal to that block's mass times the acceleration of the system.  Therefore for block B, the net force is

Consider the net forces acting on the object causing it to accelerate.

To determine the Force of Gravity in the x-direction, we must break the force of gravity into components and examine the side acting in the x-direction.  Using trigonometric functions we get that 

We know that the force of gravity is equal to mg

According to Newton’s 2nd law the force is equal to the mass times the acceleration of the object.

The force of friction is directly related to   (the coefficient of friction)  times the normal force.  In this case the normal force is equal to the y component of the force of gravity.

Therefore 

If we substitute this in our original net force equation

Notice that mass is in each piece of the equation so we can cancel it out.

We can also use our kinematic equations to determine the speed of the object at the bottom of the incline.  We can represent  to be the length of the slope.

Since the ball is assumed to be at rest at the top of the incline, the initial velocity at the top will be 0.

Therefore 

Solving for  on its own we get

In the original situation, the force of gravity is the only force pulling on the object.  Therefore the acceleration is 

We can substitute this value into our velocity equation

In the second situation the velocity is one half of the original velocity.  Therefore

We can place this back into our kinematic equation and rearrange it to solve for the acceleration.

We can then substitute that into our net force acceleration equation

Notice that g is in all of these terms so we can cancel it out.

Now we can substitute and solve

The equation for the force due to friction is , where μ is the coefficient of kinetic friction. Since there is only one force acting upon the object, the force due to friction, we can find its value using the equation . We can equate these two force equations, meaning that . We can solve for the normal force, but we need to find ma in order to find .

The problem gives us the mass of the crate, but we have to solve for the acceleration.

Start by finding the initial velocity. The problem gives us distance, final velocity, and change in time. We can use these values in the equation below to solve for the initial velocity.

Plug in our given values and solve.

We can use a linear motion equation to solve for the acceleration, using the velocity we just found. We now have the distance, time, and initial velocity.

Plug in the given values to solve for acceleration.

Now that we have the acceleration and the mass, we can return to our first equation for force.

The normal force is the same as the mass times gravity.

In this format, the masses cancel on both sides of the equation/

Now we can plug in our value for acceleration and gravity to solve for the coefficient of friction.