CK-12 People's Physics Book Version 2
Page 5
The magnitude of kinetic friction does not depend on the relative velocity or acceleration of the two objects.
Friction always points in the direction opposing motion. If the net force (not counting friction) on an object is lower than the maximum possible value of static friction, friction will be equal to the net force in magnitude and opposite in direction.
Spring Force
Finally, the last force we will cover is that exerted by a stretched spring. Any spring has some equilibrium length, and if stretched in either direction it will push or pull with a force equal to:
Spring Example
Question: A spring with a spring constant of has an uncompressed length of and a fully compressed length of . What is the force required to fully compress the spring?
Solution: We will use the equation to solve this. We simply have to plug in the known value for the spring and the distance to solve for the force.
Short Summary
An object will not change its state of motion (i.e., accelerate) unless an unbalanced force acts on it. Equal and oppositely directed forces on the same object do not produce acceleration.
The force of gravity is called weight. Near the surface of a planet, it has magnitude and is directed perpendicular to its surface. This is different from the Gravitational Constant, and differs from planet to planet.
Your mass does not change when you move to other planets --- although your weight does --- because mass is a measure of how much matter your body contains, and not how much gravitational force you feel.
To calculate the net force on an object, you need to calculate all the individual forces acting on the object and then add them as vectors.
Newton’s Third Law states for every force there is an equal but opposite reaction force. To distinguish a third law pair from merely oppositely directed pairs is difficult, but very important. Third law pairs must obey three rules: (1) Third law force pairs must be of the same type of force. (2) Third law force pairs are exerted on two different objects. (3) Third law force pairs are equal in magnitude and oppositely directed. Example: A block sits on a table. The Earth’s gravity on the block and the force of the table on the block are equal and opposite. But these are not third law pairs, because they are both on the same object and the forces are of different types. The proper third law pairs are: (1) earth’s gravity on block/block’s gravity on earth and (2) table pushes on block/ block pushes on table.
Free-Body Diagram Example
Question: Using the diagram below, find the net force on the block. The block weighs and the inclined plane has a coefficient of friction of .
Answer:
The first step to solving a Newton's Laws problem is to identify the object in question. In our case, the block on the slope is the object of interest.
Next, we need to draw a free-body diagram. To do this, we need to identify all of the forces acting on the block and their direction. The forces are friction, which acts in the negative x direction, the normal force, which acts in the positive y direction, and gravity, which acts in a combination of the negative y direction and the positive x direction. Notice that we have rotated the picture so that the majority of the forces acting on the block are along the y or x axis. This does not change the answer to the problem because the direction of the forces is still the same relative to each other. When we have determined our answer, we can simply rotate it back to the original position.
Now we need to break down gravity (the only force not along one of the axises) into its component vectors (which do follow the axises). Yet these are only the acceleration of gravity so we need to multiply them by the weight of the block to get the force.
Now that we have solved for the force of the y-component of gravity we know the normal force (they are equal). Therefore the normal force is . Now that we have the normal force and the coefficient of static friction, we can find the force of friction.
The force of static friction is greater than the component of gravity that is forcing the block down the inclined plane. Therefore the force of friction will match the force of the x-component of gravity. So the net force on the block is Therefore the net force on the block is .
Newton's Laws Problem Set
A VW Bug hits a huge truck head-on. Each vehicle was initially going . Which vehicle experiences the greater force?
Which experiences the greater acceleration?
Is it possible for me to wave my hand and keep the rest of my body perfectly still? Why or why not?
How does a rocket accelerate in space, where there is nothing to ‘push off’ against?
Is there a net force on a hammer when you hold it steady above the ground? If you let the hammer drop, what’s the net force on the hammer while it is falling to the ground?
If an object is moving at constant velocity or at rest, what is the minimum number of forces acting on it (other than zero)?
If an object is accelerating, what is the minimum number of forces acting on it?
You are standing on a bathroom scale. Can you reduce your weight by pulling up on your shoes? (Try it.)
When pulling a paper towel from a paper towel roll, why is a quick jerk more effective than a slow pull?
You and your friend are standing on identical skateboards with an industrial-strength compressed spring in between you. After the spring is released, it falls straight to the ground and the two of you fly apart. If you have identical masses, who travels farther?
If your friend has a bigger mass who goes farther?
If your friend has a bigger mass who feels the larger force?
If you guys have identical masses, even if you push on the spring, why isn’t it possible to go further than your friend?
Explain the normal force in terms of the microscopic forces between molecules in a surface.
A stone with a mass of is sitting on the ground, not moving. What is the weight of the stone?
What is the normal force acting on the stone?
The stone from the last question is now being pulled horizontally along the ground at constant speed in the positive direction. Is there a net force on the stone?
A spring with spring constant has an uncompressed length of . When fully compressed, it has a length of . What force is required to fully compress the spring?
Measuring velocity is hard: for instance, can you tell how fast you’re going around the Sun right now? Measuring acceleration is comparatively easy — you can feel accelerations. Here’s a clever way to determine your acceleration. As you accelerate your car on a flat stretch, you notice that the fuzzy dice hanging from your rearview mirror are no longer hanging straight up and down. In fact, they are making a angle with respect to the vertical. What is your acceleration? (Hint: Draw a FBD. Consider both and equations.)
Draw free body diagrams (FBDs) for all of the following objects involved (in bold) and label all the forces appropriately. Make sure the lengths of the vectors in your FBDs are proportional to the strength of the force: smaller forces get shorter arrows! A man stands in an elevator that is accelerating upward at .
A boy is dragging a sled at a constant speed. The boy is pulling the sled with a rope at a angle.
Your foot presses against the ground as you walk.
The picture shown here is attached to the ceiling by three wires.
Analyze the situation shown here with a big kid pulling a little kid in a wagon. You’ll notice that there are a lot of different forces acting on the system. Let’s think about what happens the moment the sled begins to move.
First, draw the free body diagram of the big kid. Include all the forces you can think of, including friction. Then do the same for the little kid.
Identify all third law pairs. Decide which forces act on the two body system and which are extraneous.
Explain what conditions would make it possible for the two-body system to move forward.
Break the force vector on the right into its and components, and .
For both figur
es below, find the net force and its direction (i.e., the magnitude of and the angle it makes with the axis). Draw in
Andreas and Kaya are pulling a wagon. Andreas is pulling with a force of towards the northeast. Kaya is pulling with a force of towards the southeast. The wagon has a mass of . What is the acceleration and direction of motion of the wagon?
Laura and Alan are pulling a wagon. Laura is pulling with a force of towards the northeast. Alan is pulling with a force of directly east. The wagon has a mass of . What is the acceleration and direction of motion of the wagon?
When the box shown below is pulled with a force of , it just starts to move (i.e., the maximum value of static friction is overcome with a force of ). What is the value of the coefficient of static friction, ?
A different box, this time in mass, is being pulled with a force of and is sliding with an acceleration of . Find the coefficient of kinetic friction, .
The man is hanging from a rope wrapped around a pulley and attached to both of his shoulders. The pulley is fixed to the wall. The rope is designed to hold of weight; at higher tension, it will break. Let’s say he has a mass of . Draw a free body diagram and explain (using Newton’s Laws) whether or not the rope will break
Now the man ties one end of the rope to the ground and is held up by the other. Does the rope break in this situation? What precisely is the difference between this problem and the one before?
For a boy who weighs on Earth what are his mass and weight on the moon (where ?
A woman of mass weighs herself in an elevator.
If she wants to weigh less, should she weigh herself when accelerating upward or downward?
When the elevator is not accelerating, what does the scale read (i.e., what is the normal force that the scale exerts on the woman)?
When the elevator is accelerating upward at , what does the scale read?
A crane is lowering a box of mass with an acceleration of . Find the tension in the cable.
If the crane lowers the box at a constant speed, what is the tension in the cable?
The large box on the table is and is connected via a rope and pulley to a smaller box, which is hanging. The mass is the highest mass you can hang without moving the box on the table. Find the coefficient of static friction .
Find the mass of the painting. The tension in the leftmost rope is , in the middle rope it is , and in the rightmost rope it is .
Find Brittany’s acceleration down the frictionless waterslide in terms of her mass , the angle of the incline, and the acceleration of gravity .
The physics professor holds an eraser up against a wall by pushing it directly against the wall with a completely horizontal force of . The eraser has a mass of . The wall has coefficients of friction and Draw a free body diagram for the eraser.
What is the normal force acting on the eraser?
What is the frictional force equal to?
What is the maximum mass m the eraser could have and still not fall down?
What would happen if the wall and eraser were both frictionless?
A tractor of mass accelerates up a incline from rest to a speed of in . For all of answers below, provide a magnitude and a direction.
What net force has been applied to the tractor?
What is the normal force, on the tractor?
What is the force of gravity on the tractor?
What force has been applied to the tractor so that it moves uphill?
What is the source of this force?
A heavy box (mass ) is dragged along the floor by a kid at a angle to the horizontal with a force of (which is the maximum force the kid can apply).
Draw the free body diagram.
What is the normal force ?
Does the normal force decrease or increase as the angle of pull increases? Explain.
Assuming no friction, what is the acceleration of the box?
Assuming it begins at rest, what is its speed after ten seconds?
Is it possible for the kid to lift the box by pulling straight up on the rope?
In the absence of friction, what is the net force in the direction if the kid pulls at a angle?
In the absence of friction, what is the net force in the direction if the kid pulls at a angle?
In the absence of friction, what is the net force in the direction if the kid pulls at a angle?
The kid pulls the box at constant velocity at an angle of . What is the coefficient of kinetic friction between the box and the floor?
The kid pulls the box at an angle of , producing an acceleration of . What is the coefficient of kinetic friction between the box and the floor?
For the following situation, identify the law force pairs on the associated free body diagrams. Label each member of one pair each member of the next pair and so on. The spring is stretched so that it is pulling the block of wood to the right.
Draw free body diagrams for the situation below. Notice that we are pulling the bottom block out from beneath the top block. There is friction between the blocks! After you have drawn your FBDs, identify the law force pairs, as above.
Spinal implant problem — this is a real life bio-med engineering problem!
Here’s the situation: both springs are compressed by an amount . The rod of length is fixed to both the top plate and the bottom plate. The two springs, each with spring constant , are wrapped around the rod on both sides of the middle plate, but are free to move because they are not attached to the rod or the plates. The middle plate has negligible mass, and is constrained in its motion by the compression forces of the top and bottom springs.
The medical implementation of this device is to screw the top plate to one vertebrae and the middle plate to the vertebrae directly below. The bottom plate is suspended in space. Instead of fusing broken vertebrates together, this implant allows movement somewhat analogous to the natural movement of functioning vertebrae. Below you will do the exact calculations that an engineer did to get this device patented and available for use at hospitals.
Find the force, , on the middle plate for the region of its movement . Give your answer in terms of the constants given. (Hint: In this region both springs are providing opposite compression forces.)
Find the force, , on the middle plate for the region of its movement . Give your answer in terms of the constants given. (Hint: In this region, only one spring is in contact with the middle plate.)
Graph vs. . Label the values for force for the transition region in terms of the constants given.
You design a mechanism for lifting boxes up an inclined plane by using a vertically hanging mass to pull them, as shown in the figure below.
The pulley at the top of the incline is massless and frictionless. The larger mass, , is accelerating downward with a measured acceleration a. The smaller masses are and ; the angle of the incline is , and the coefficient of kinetic friction between each of the masses and the incline has been measured and determined to be .
Draw free body diagrams for each of the three masses.
Calculate the magnitude of the frictional force on each of the smaller masses in terms of the given quantities.
Calculate the net force on the hanging mass in terms of the given quantities.
Calculate the magnitudes of the two tension forces and in terms of the given quantities.
Design and state a strategy for solving for how long it will take the larger mass to hit the ground, assuming at this moment it is at a height above the ground. Do not attempt to solve this: simply state the strategy for solving it.
You build a device for lifting objects, as shown below. A rope is attached to the ceiling and two masses are allowed to hang from it. The end of the rope passes around a pulley (right) where you can pull it downward to lift the two objects upward. The angles of the ropes, measured with respect to the vertical, are shown. Assume the bodies are at rest initially.
Suppose you are able to measure the masses and of the two hanging objects as well as the tension . Do you then have
enough information to determine the other two tensions, and ? Explain your reasoning.
If you only knew the tensions and , would you have enough information to determine the masses and ? If so, write and in terms of and . If not, what further information would you require?
A stunt driver is approaching a cliff at very high speed. Sensors in his car have measured the acceleration and velocity of the car, as well as all forces acting on it, for various times. The driver’s motion can be broken down into the following steps: Step 1: The driver, beginning at rest, accelerates his car on a horizontal road for ten seconds. Sensors show that there is a force in the direction of motion of , but additional forces acting in the opposite direction with magnitude . The mass of the car is .
Step 2: Approaching the cliff, the driver takes his foot off of the gas pedal (There is no further force in the direction of motion.) and brakes, increasing the force opposing motion from to . This continues for five seconds until he reaches the cliff.
Step 3: The driver flies off the cliff, which is high and begins projectile motion.
Ignoring air resistance, how long is the stunt driver in the air?
For Step 1: Draw a free body diagram, naming all the forces on the car.
Calculate the magnitude of the net force.
Find the change in velocity over the stated time period.
Make a graph of velocity in the direction vs. time over the stated time period.
Calculate the distance the driver covered in the stated time period. Do this by finding the area under the curve in your graph of (iv). Then, check your result by using the equations for kinematics.