AP Physics 1 Dynamics
Explore AP Dynamics physics principles and extensions. Begin with Newton's Laws of Motion and forces relative to multiple cart systems.
Newton’s First Law: Law of Inertia
Law of inertia: An object at rest will remain at rest and an object in motion will stay in motion unless it encounters an external net force.
The more mass the more a system resists acceleration when an external net force is present.
How does the inertia of each system of cart(s) compare?
Inertia is directly related to mass.
Here we have a system consisting of:
- A 500 kg green locomotive cart.
- A 500 kg green locomotive and 300 kg blue cart.
- A 500 kg green locomotive, 300 kg blue, and 250 kg red cart.
- A 500 kg green locomotive, 300 kg blue, and 250 kg red, and 450 kg yellow cart.
How may times more massive?
How many time more massive is something that will be important for ratio problems, also called rule of ones, that can be used throughout physics. Review the link if you want more or a refresher.
Ratio of new mass to original:
(New Mass)/(Original Mass)
The original cart I will make a comparison to is the 500 kg green locomotive cart. It of course will be one times the original seen here.
(Green Cart)/(Original Mass) = (500 kg)/(500 kg) = 1 times
When the systems includes multiple carts pulled by the green locomotive, the system will have a multiple of the original green cart's mass.
(Green+Blue)/(Original Mass) = (800 kg)/(500 kg) = 1.6 times
(Green+Blue+Red)/(Original Mass) = (1050 kg)/(500 kg) = 2.1 times
(Green+Blue+Red+Yellow)/(Original Mass) = (1500 kg)/(500 kg) = 3 times
Equilibrium: No Net Force External to the System
When there is no external overall net force the system is in equilibrium. The center of mass of the system will be at rest or in constant motion in a straight line if already moving.
Depicting Motion
Motion is depicted many ways in problems.
One way of showing constant motion (dynamic equilibrium) is with a motion diagram.
Each dot or arrow is placed in equal time intervals. There is the same displacement each time interval. Constant motion is shown with arrows of the same length in the same direction per interval. Dots equally spaced represent the same but direction is unknown unless they are numbered. Observe how the animation shows each displacement in equal time intervals for constant velocity.
In these motion diagrams you see acceleration. Notice how the length of the displacement vectors increase per time interval.
Not in Equilibrium: Acceleration
When there is an overall net external force the system is not in equilibrium and accelerating. The center of mass of the system will be accelerating.
Newton’s Second Law: Fnet = ma
An unbalanced net force will create an acceleration. The amount of acceleration will depend on its mass.
Equation: Fnet = ma
Rearranged to analyze acceleration: 
How will acceleration of the cart compare when the green locomotive is pulling a system of a different number of carts?
The net force will be the same in this example or 1x the original.
How many times the original green carts mass
- Green: 1 Times Original Mass
- Green & Blue: 1.6 Times Original Mass
- Green, Blue, & Red: 2.1 Times Original Mass
- Green, Blue, Red, & Yellow: 3 Times Original Mass
Acceleration that results from the same force but various times the mass
Set up a ratio of new Fnet/m divided by the original a = Fnet/m to determine how many times the acceleration will be different.
A) Green & Blue acceleration with 1.6 times the original mass and the same force applied
B) Green, Blue, and Red acceleration with 2.1 times the original mass and the same force applied
C) Green, Blue, Red and Yellow acceleration with 3 times the original mass and the same force applied
Finding acceleration directly plugging numbers into the equation.
What is the acceleration of the 500 kg green if its engine applies 2000 N forward?
a = 2000/500 = 4 m/s2 Forward
What is the acceleration of the 800 kg green and blue cart train if the green engine applies 2000 N forward?
Using a ratio to solve this problem. We determined the ratio to 0.625 times the acceleration.
- (Green alone acceleration) x (acceleration ratio of green & blue)
- (4 m/s2)(0.625) = 2.5 m/s2 Forward
Using a direct calculation you would get the same result.
- a = Fnet/m
- a = 2000N/800kg
- a = 2.5 m/s2 Forward
Question: Determine the acceleration of the green engine pulling the green, red, and blue together.
Question: Determine the acceleration of the green engine pulling the green, red, blue, and yellow together.
Here are the resulting differences in acceleration and what they would look like relatively when the green cart is pushing itself or a combination of multiple carts will less acceleration because of more mass and more inertia.
Newton’s Third Law
Forces come in equal and opposite action reaction pairs. When object A applies a force on object B, object B applies an equal and opposite force of object A.
As the cart’s wheels apply a force backward to the ground, the ground applies an equal and opposite force to the cart propelling it forward.
Newton's Third Law also applies on the force interaction between each cart.
First, notice that the whole group of carts are accelerating at the same rate calculated earlier to be 1.333 m/s2.
The force interaction will be different in magnitude between carts. Since the further back the interaction, the less mass has to be pulled to get the same acceleration of 1.333 m/s2 Forward.
It requires less force to accelerate less mass at the same acceleration.
But between each cart the force will be equal and opposite.
The free body diagram above and below the cart show the force and direction on each cart.
Calculating the Amount of Force on Each Cart
The green carts engine applies 2000N of force to accelerate the 1500kg train. The combined the train is accelerating at 1.333 m/s2 forward together.
How much force is applied between the green and blue cart?
Fnet = ma
The blue cart must be pulled with enough force to accelerate the combined blue, red, and yellow cart.
(300+250+450) = 1000 kg
Fnet = (1000)(1.333) = 1333 N Forward
Because of Newton's Third Law of Motion, the blue cart will apply an equal and opposite force on the green cart.
How much force is applied to the blue and red cart?
- The red cart must be pulled with enough force to accelerate the combined red, and yellow cart.
- (250+450) = 700 kg
- Fnet = (700)(1.333) = 933.1 N Forward
- Because of Newton's Third Law of Motion, the red cart will apply an equal and opposite force on the blue cart.
How much force is applied between the red and yellow cart?
- The yellow cart must be pulled with enough force to accelerate the mass of the 450 kg yellow cart.
- Fnet = (450)(1.333) = 599.85 N Forward
- Because of Newton's Third Law of Motion, the yellow cart will apply an equal and opposite force on the red cart.
Links
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- Back to the Main Forces Page
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