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Our Balanced Force model model is the first of two units in Dynamics. The purpose of Dynamics is to explain why objects move as they do. Dynamics can be challenging because many people have incorrect ideas about why objects move the way they do.
A force is a push or pull between two objects. Forces explain motion. Forces should always be labeled using on/by notation like: Fn on the cart by the ground. Forces are vectors.
Contact vs. Non-Contact Forces
Contact: forces that must involve touching objects.
Non-Contact: forces between objects that need not touch to exert forces between them.
Every mass pulls every other mass in the universe towards it. When Earth pulls on objects around here, the force is large because the mass of the Earth is so large.
The free body diagram is the first step to understanding why an object either moves uniformly or non-uniformly. To create a free body diagram, first choose a system. A system is what objects you are analyzing. Then represent the object as a dot. This is similar to what we did with motion maps. Next draw an arrow which starts at the dot and points in the direction of the push/pull for each force acting on the object. Remember to label your arrow with the type of force and what object the force is acting on and what object is exerting the force.
A checklist for drawing free body diagrams:
If the object is on a planet then there will be a gravitational force pointing towards the center of the planet.
Each object/surface that comes in contact with the object you are drawing a free body diagram for will exert a normal force on your object. This force will be perpendicular to the line where the objects touch.
Each string/rope etc. that comes in contact with your object you are drawing a free body diagram for will exert a tension force (pull). This force will be parallel to the string/rope.
If the object is moving over a surface, or is at rest but could move, there may be a friction force. Friction forces always point in the opposite direction of velocity or intended velocity (i.e. which way it would move if it started moving).
If you can't identify the type of force or what object is causing the force it's extremely likely you are trying to invent a bogus force! Finally make sure the size of your vectors are correct and hash any equal arrows.
Example 1:
Example 2:
A gift box is sitting on the surface. Draw the free body diagram for the box & write a summation equation. (The bow on top does not add any weight, it's just for show) :)
Explanation:
There is a normal force on the box by the table because the gift box is touching the table. This force goes perpendicular to the table & it's a push - so up. You will always have to specify the gravitational force, unless the location is space. The two hash marks indicate that the forces are equal. When drawing a free body diagram hashes are necessary if any forces are equal.
Summation Equation:
∑Fy=Fn+Fg=0
The sum of the forces will equal 0 because the box is at rest therefore it has a constant velocity of 0, so the net force is 0 N by Newton's 1st Law.
Example 3:
A box sits motionless on a ramp. Draw a freebody diagram for the box.
Explanation:
Fn on the box by the ramp is drawn diagonally because the Fn needs to be perpendicular to the surface. The Earth's gravity still pulls straight down. If the box were to move, it wide slide down the ramp, so friction points up the ramp to oppose that.
Example 4:
A person pushes a giant bowling ball on a friction-less surface. The ball speeds up.
The gravitational force from Earth applies a downward force on the bowling ball and the floor applies a normal upward force onto the bowling ball. They are equal because the forces are balanced (Newton's 1st Law). Meaning, the bowling ball is not accelerating in the vertical direction. Then there is the normal force on the bowling ball by the person because of the person pushing the bowling ball. The horizontal forces are unbalanced so the ball must speed up.
Box A is hanging from the ceiling through two ropes. Draw a freebody diagram and find the Ft on both ropes B and C if the mass of the box is 2kg and the tension on the ropes is the same. (Assume that this is on Earth)
Explanation: Because the mass of the box is 2kg, we can determine that the overall tension on BOTH ropes is 20N. Which means, that the tension on each is 10N. We can determine this because of the equation: Fg(Force of gravity) = m(mass) x g(gravitational field)
Because we know the mass of the box, and the gravitational field, we can successfully figure out the Fg on A.
(2kg- Remember to always put the mass in kg, but in this case, it is given as kg) x (The gravitational field on Earth, which is -10 N/kg) = -20N
Why is this number negative? Well, just think of the weight of an object as the amount the Earth is pulling down on the object. Because the Earth is pulling down on the object, the Fg will always be negative on Earth. The Ft ,on the other hand, is simply the Fg x -1. Because the object is not moving (uniform motion), then the forces must be balanced. This is Newton's 1st Law.
Remember that we have to divide the overall force on both by two because there are two ropes holding up the box and the question asks for the individual force on each rope.
Before writing a summation equation (or attempting any other force problem), ALWAYS draw a force diagram first. After you have your force diagrams drawn out: imagine your force diagram as a coordinate plane. The forces going in the horizontal direction would be on the x-axis, and the vertical forces on the y-axis. In physics, what happens horizontally is totally separate from what happens vertically. Therefore, the forces going left and right will be in the x direction, and the ones going up and down in the y direction.
To create a summation equation, start with two equations (though there may not necessarily be forces in both directions) ∑Fx = ∑Fy =
If there are no forces acting in one of the directions, simply make the equation equal to 0N. For example: ∑Fx = 0N.
In most cases, when there are forces acting in the direction, your equation should display all of the forces being added up. If you have a balanced force situation (meaning that the object(s) are moving at a constant velocity the entire time), then that particular direction's summation equation should add up to 0N. This is because of Newton's Second Law, which states that ∑F=ma (mass x acceleration). If the acceleration is 0, then the net force must be 0N.
Example 1:
If you have both a tension force (Ft) and a friction force (Ff) and the object being pulled is moving at a constant velocity, then the summation equation for the x direction would look as follows:
∑Fx = Ft on object by rope + Ff on object by ground = 0N.
This works exactly the same way for gravitational forces (Fg) and normal forces (Fn). If the object is not accelerating:
∑Fy = Fg on object by Earth + Fn on object by ground = 0N.
However, there are some occasions where there are unbalanced force models (object is accelerating), and the summation equations would not add up to 0N. This is also due to Newton's Second Law. When the acceleration is positive, then the net force will be positive. Likewise, when the acceleration is negative, the net force will also have to be negative. Summation equations are especially useful when doing calculations with lots of forces acting on the object.
An object in motion will continue to move uniformly unless acted upon by an unbalanced force.
Three notes about this definition:
1. This definition does not explicitly address objects at rest. However, an object at rest is always moving at 0 m/s so that's uniform motion.
2. This means that if an object moves uniformly, then the forces on it must be balanced (i.e. cancel out, or sum to zero).
3. If an object speeds up or slows down then the forces must be unbalanced.
New term:
Inertia is an object's resistance to acceleration--it will continue moving in a straight line unless acted upon by an unbalanced force. This is another way of thinking about Newton's first law.
Inertia is not a force. It never goes on a free body diagram.
Every object with mass has inertia.
Example 1:
The dry ice in this video moves in a straight line at a constant velocity even though no forces act on it! The dry ice is interesting because it moves on the table with almost no friction since every side of the ice is turning straight from a solid into a gas (so it makes it's own gas cushion as it goes). If friction was this small for most object's it would be really easy to understand Newton's 1st Law!
Example 2:
Consider a marble going down a friction-less track, another one going up a friction-less track and a third marble on a completely flat friction-less track.
In the first example, the normal force is titled a little bit to the right since it's perpendicular to the surface. Since there is a little forces to the right, but nothing to the left the horizontal forces are unbalanced and the marble speeds up (i.e. moves non-uniformly).
In the second case, the normal force is titled a little bit to the left. Since there are some forces leftwards, but no forces rightwards the horizontal forces are again unbalanced and the marble once again moves non-uniformly. In this case the marble slows down since the extra force is in the opposite direction as the velocity. So the velocity and acceleration are opposite and the object slows down.
If a marble were on an endless ramp parallel to the ground, smooth enough that there is no friction, it will continue to roll forever, at the same speed, unless acted upon by an unbalanced force. This is demonstrated in the third diagram, as opposed to the first two, where the object is acted upon by unbalanced forces, causing a change in velocity.
Example 3:
We see evidence of Newton's First Law in everyday life as well. When riding in a car that stops abruptly, the passengers are thrown forward - especially if they are not wearing a seat belt. Why? Because they were in motion just a moment ago, and as Newton's First Law dictates, they naturally tend to stay in motion unless acted on by an unbalanced force. Without a seat belt there is no force to slow the person down, so they don't slow down. The car's slows down because there are unbalanced forces (i.e. friction with the road) to slow it down. The fact that the person resists acceleration (or maintains a constant velocity) is called inertia.
Example 4:
Draw a free body diagram for a block sliding down an incline at a constant velocity.
This picture shows a free-body diagram for the block on the ramp. Note, for the block to move down the ramp at a constant speed the forces in the direction of the ramp must be balanced. This means there MUST be friction.
The force that object A exerts on object B is equal in magnitude and opposite in direction to the force that object B exerts on object A.
F(on A by B) = -F(on B by A)
Newton's 3rd Law says that forces always come in pairs. If a dump truck pushes on me then I must also push on the truck with the same amount of force. If the size of one of these forces change they must both change together. The direction of these two "force pairs" is always opposite. One misconception people have is that bigger objects exert bigger forces on smaller objects. However, by Newton's 3rd Law, the bigger object must push on the smaller object with exactly as much force as the smaller object exerts on the bigger one.
Things to remember about Newton's 3rd Law Force Pairs:
Note that both force pairs can never show up on the same free body diagram, as they can never be exerting a force on the same object. By definition one force is on object "A" while the other is on object "B."
When you do the on/by notation there must not be more then 2 objects stated in the on by notation for both force pairs. For example, A on B pairs with B on A, but A on B does not pair with A on C.
In addition, Force Pairs can never be two different types of forces. For example, a Normal Force (Fn) cannot have a force pair that is a Gravitational Force (Fg) or Friction Force (Ff). A normal force must pair with another Normal Force.
A note on "Action-Reaction"
A common definition of Newton's 3rd Law is: "For every action, there is an equal and opposite reaction." There is nothing wrong with this definition, but the problem is that the vast majority of people who have memorized this definition have no clue what it means. Unfortunately this definition is not easily understood which is why we don't use it.
Example 1:
A bus is moving at a constant velocity of 200 miles per hour and a fly is flying towards it at a constant velocity of 1 mile per hour. How does the size of the force on the bus by the fly compare to the size of the force on the fly by the bus?
Upon collision, the two objects exert the same amount of force on one another because of Newton's Third Law (Force on the bus by fly = Force on the fly by bus). After they collide, the bus will have a very small acceleration and the fly will face a huge acceleration. This is because the inertia of the bus is much larger than the inertia of the fly. After the collision, the remains of the fly will travel at the same speed as the bus.
You may consider this as seeming a bit off, but it's true! How do we explain this? One thing we must do is separate the cause (i.e. force) from the effect (i.e. acceleration). People get confused because the acceleration of the bus is so different then the acceleration of the fly. The reason the accelerations are so different is that the fly has such little mass that it offers very little inertia (i.e. the resistance to acceleration). For example, if you exert the same force on a dump truck and a pencil the effect will not be the same! So even though the effect is quite different the cause is the same.
Problem Solving Approach for Dynamics Problems
Question: find the magnitude (size) of all the forces acting on a 4 kg ball sitting at rest.
Tip: this problem solving approach might be overkill for a simple question like this. On a harder question it makes a world of a difference though!
You need to figure out what object(s) you will need to analyze in order to answer your question. This is important because forces on objects in your system by objects outside your system (i.e. environment) show up on a free body diagram. However, forces on one object in your system by another object in your system can be ignored since both of the force pairs act on objects within your system. These internal forces need not be drawn on free body diagrams.
In this case the system is just the ball. Even though different atoms in the ball exert forces on each other we can ignore these forces because they are on one part of the system by another part of the system. However a force from something outside the system, like Earth, must be considered. In this problem the system is fairly obvious, but in harder problems with multiple object's choosing a system which is easy to analyze is more challenging.
The forces present that are acting on the ball are the following: Fg on ball by Earth, and Fn on ball by table. The tick marks on the diagram represent the fact that the two forces have the same magnitude, as the ball is at rest. There must be a normal force to balance the gravitational force, since the vertical motion of the object is uniform. There are no horizontal forces acting on the ball.
There are no horizontal (x) forces, therefore, the horizontal net force is 0 N. When you input numbers, the gravitational force is going down and thus will be negative. The forces that are present on the object, Fg, and Fn, are balanced. This is why the object is moving uniformly (i.e. at rest - which is a constant velocity). When objects are balanced their total net force equals 0.
Tip: don't forget about signs. Forgetting about direction is a very common error!
Step 4: Figure out the magnitudes of any forces you know
Fg=mg= (4kg)(-10N/kg)= -40 N
Step 5: Determine the remaining forces
Since the normal force is the same magnitude but opposite direction as the gravitational force, Fn = 40 N.
Ex: 40N - Fn = 0
-Fn = -40N
Fn = 40N
Step 6: Check your work
For any quantitative (i.e. with numbers) problems during the Balanced Force Model unit the forces will always balance! This means means the object will either be at rest or move constantly but with NO acceleration. If this is true for your object, show that the horizontal/vertical forces being acted upon are equal in retrospect by including hash marks. If the information of your object does not match up with your model, check again to see what type of motion the object is undergoing.
Ranking forces in a free body diagram
An empty fishbowl has been placed on a stationary table. The fishbowl is more massive then the table. Rank the sizes of all the forces acting on both objects.
List of forces:
- Fn on fishbowl by pedestal(table)
- Fg on fishbowl by Earth
- Fn on table by floor
- Fn on table by fishbowl
- Fg on table by Earth
Let's figure out what we know:
- We know that the forces on the fish bowl balance.
- There are only two forces acting on the fishbowl which is Fn on fishbowl by table and Fg on fishbowl by Earth
- We know these two forces are equal because the fishbowl is stationary and so is the table, therefore, the forces are balanced on both objects.
- Conclusion: Fn on fishbowl by table = Fg on fishbowl by Earth
- There are three forces acting on the table which are: Fn on table by floor, Fn on table by fishbowl, and Fg on table by Earth
- We know that the Fg and the Fn on table by fishbowl added together are equivalent to the Fn on table by floor
- We know this because the forces are balanced and the table is stationary
- We know that the Fn on table by floor is greater than the Fn on table by fishbowl and the Fg on table by fishbowl because the Fn on table by fishbowl and the Fg both add up together and equal Fn on table by floor
- We know that Fn on fishbowl by table and Fn on table by fishbowl are equal by Newton's 3rd Law (Fon A by B = -Fon B by A.)
-Therefore Fn on fishbowl by table = Fn on table by fishbowl = Fg on fishbowl by Earth
- Fn on table by fishbowl is greater than Fg on table by Earth because we know that Fg on table by Earth is less then Fg on fishbowl by Earth (since the fish bowl is more massive) and we also know that Fn on table by fishbowl = Fg on fishbowl by Earth.
Conclusion: Fn on table by floor> Fn on fishbowl by table =Fg on fishbowl by earth= Fn on table by fishbowl >Fg on table by earth
How can any object accelerate if every object has another force which is equal in size, but opposite in direction?
Challenging Free Body Diagram Problem
Draw free body diagrams for blocks A and B. Block A is more massive then B. Both blocks are at rest. Remember to mark any forces which are the same size.
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