The component of  perpendicular to the slope is , where  is the angle between the ground and the incline.

 where  is the normal force.

The force of friction, which is in the direction opposing motion is:

 is the coefficient of kinetic friction.

The component of  parallel to the slope is:

The net force on the block is:

Begin by diagraming the forces acting on the mass in the problem:

The mass itself creates a force due to gravity in the downward direction:

For the mass to remain stationary, the forces must be in equilibrium. Therefore the sum of forces in the x and y directions each must be zero:

The x and y tensions can be written in terms of the magnitude of the tension in each cable. Begin with the x direction:

Because the angle is the same for each, the tension in each angle must be equivalent. Use this property when performing the force balance in the y direction:

Begin by drawing a force diagram of forces acting on the mass:

The mass itself creates a force due to gravity in the downward direction.

For the mass to remain stationary, the forces must be in equilibrium. Therefore the sum of forces in the x and y directions each must be zero:

The x and y tensions can be written in terms of the magnitude of the tension in each cable. Begin with the x direction:

Because the angle is the same for each, the tension in each angle must be equivalent. Use this property when performing the force balance in the y direction:

Begin by diagraming the forces acting on the mass:

The mass itself creates a force due to gravity in the downward direction.

For the mass to remain stationary, the forces must be in equilibrium. Therefore the sum of forces in the x and y directions each must be zero:

The x and y tensions can be written in terms of the magnitude of the tension in each cable. Begin with the x direction:

Because the angle is the same for each, the tension in each angle must be equivalent. Use this property when performing the force balance in the y direction:

During terminal velocity:

Thus, by Newton's second law:

The free body diagram of the block is given above. This block has three forces acting on it. First, it's weight under the influence of gravity, which is given as . Second, the normal force of the plane, which is given as . Third, the friction force, which acts opposite to its direction of motion and is given as . We choose a coordinate system so that our x-axis aligns with the motion of the block down the plane, and the y-axis aligns with the direction of the normal force. Thus the friction force points in the negative direction of the x-axis, and the normal force is aligned with the positive direction of the y-axis. However, the weight  is not along either of these axes, so we resolve the  force into its components,  along the negative y-axis, and  along the positive x-axis. 

Now we can use Newton's 2nd law to relate the given forces above. Newton's 2nd law gives us two equations: 

 and 

Because the block is constrained to move along the surface of the inclined plane, there should be no acceleration in the y direction, and so . Also, because the block moves at constant velocity down the plane, Newton's 1st law assures us that there is no acceleration in the x direction as well, therefore . Plugging these accelerations in, we find that  and 

Summing all the forces in the x-direction gives us 

Summing all the forces in the y-direction gives us

Plugging these values into the force equations above gives us the following equations:

Solving for  in the second equation gives us . Thus the normal force is equal to the cosine component of the weight. Substituting  in for  in the first equation will give us the following:

Now we solve the equation for . Adding  to each side gives us: 

Now we divide each side by  to obtain:

The final result is obtained by canceling the  factor and using the triginometric identity:

Therefore we arrive at the conclusion that 

In this question, we're told that an object of a given mass is free-falling in the air with no air resistance. We're asked to determine the amount of upward force necessary to cause this object's velocity to become constant.

In order for an object's velocity to be maintained at a constant value, that object must not be accelerating. Therefore, we'll need to determine what force will allow this object to have a net acceleration of zero.

Looking at the falling object's motion in the downward direction, we know that it is falling due to the influence of gravity. Since we know its mass, we can calculate this downward force:

We've stated that in order to have a constant velocity, we need to have a net acceleration of zero. And since we know that the object is accelerating by  in the downward direction, we will need it to accelerate upward by  to cancel it to zero.

To conclude, we'll have to match the downward force by presenting an equal upward force in order to cancel out the object's acceleration and bring its velocity to a constant value.

Imagine two men of equal strength pulling on ropes that are connected to a crate. In order to obtain the largest resulting force the two men should pull in the same direction at . Pulling on opposite ends at  would result in zero resultant force and anything besides  would cause the force to act in an unneeded direction.

Using definition of force and superposition of Forces

Each person is apply the same amount of force against gravity. The car is holding still so it has no acceleration.

Solving for and plugging in values, remembering that gravity points downwards and thus is negative.

Using

Converting to and plugging in values.