Fundamentals of Force and Newton's Laws - AP Physics 1

There are two relevant forces acting on the block in this scenario: gravity and friction. We can use Newton's second law to solve this problem:

Substituting in expressions for each force, we get:

Eliminating mass and rearranging for , we get:

At this point, we can plug in values for each variable and solve:

Start by drawing in the forces acting on each block. You could also draw in the force of gravity and the normal force for each block, but they have been omitted from the image because they cancel each other out for each block and because there is no friction in this problem.

We are given in the question that the force is 100 N. Since the blocks are connected by a string, they will therefore accelerate at the same rate, and we can treat them as a system that moves as if it were one object of total mass 80 kg (30 kg plus 50 kg). Use Newton's second law:

In this problem, the two tension forces form an action/reaction pair and therefore are equal in magnitude but opposite in direction (Newton's third law). So:

We can solve for acceleration, since the tensions cancel out.

Now that we have acceleration, we need to write a new equation in which the tension force does not cancel out so that we can solve for the tension .

Do this by using Newton's second law again, except for only one of the blocks:

Lets consider the 30 kg block. The only force acting on the 30 kg block is the tension , and the acceleration is what we found above.

The normal force is perpendicular to the plane:

First, we need to find .

We can solve for using the trigonometric equation that applies in this instance. We know the length of the side opposite of (5 m) and the length of the side adjacent to (10 m), so we can use the following equation to solve for :

Rearranging to solve this equation for , you get

Substituting in the side lengths of the given triangle, we can solve for .

Note that the normal force is one of the legs of another right triangle. The other leg is the parallel force, and the hypotenuse is the force of gravity.

Using trigonometry, we know that

because , or, in terms of this problem, .

Substituting in the known values into this equation, we can solve for the normal force:

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:

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.

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

During terminal velocity:

Thus, by Newton's second law:

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.

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.

In the question stem, we're told that two forces of equal magnitude are pointing south and the other east. We're then asked to find which way the third force must be oriented in order to cancel out the other two forces. To figure this out, it's best to draw a force diagram.

Looking at this force diagram, we actually don't even need to do any math in order to realize that the resultant force from these two forces will point in the southeast direction. Consequently, the third force will need to point in the opposite direction in order to balance things out; the third force must be oriented in the northwest direction.

As can be seen in the diagram, the force pushing down the ramp is equal to

The force pushing the object into the ramp is

The force into the ramp will be equal to the normal force, thus

The net force pushing the woman down the ramp is

At the moment she starts slipping, the net force is equal to zero

Solving for :

Solving for

Plugging in values

Because the block is resting on an inclined surface, only a component of the gravitational force will contribute to the normal force between block and ramp. It can be assumed that the only relevant force acting on this block is due to gravity.

Since the normal force only acts in the direction perpendicular to the contact surface, the correct gravitational force component to take is the perpendicular component. This gives . By plugging in the known values for , and , we get

The tensions in both ropes are caused by the bucket being pulled down by gravity, hence the only relevant forces acting in this scenario are those in the y-direction. Since the bucket is being held up against gravity, the upward forces should balance exactly with the downward forces. The forces in the y-direction can be balanced like this:

, where is the tension in rope 1. Thus we find that: