Closed Systems vs. Open Systems

A closed system refers to a system that doesn’t lose mass, energy, charge, etc so conserved quantities are considered to be constant. On the other hand, an open system refers to exchanges of energy, charge, etc with the object(s) surroundings. 

Open systems may seem to violate the conservation of mass, charge, matter, and energy since the amount of the previously stated quantities can increase or decrease without replacement.  

Here are some key points about closed and open systems in the context of momentum:

• In a closed system, the total momentum of all objects within the system is conserved. This means that the total momentum of the system before and after an interaction remains the same.
• In an open system, the total momentum of all objects within the system is not necessarily conserved. This is because an open system can exchange momentum with objects outside of the system.
• In an open system, the total momentum of the system can be calculated by taking into account the momentum of all objects within the system as well as the momentum of objects outside the system that interact with the system.
• In an open system, the total momentum of the system may change due to the transfer of momentum between the system and objects outside the system.
• In order to analyze the momentum of an open system, it is important to consider all interactions with objects outside the system and how they affect the total momentum of the system.

Image Credit: physics.usyd.edu

Momentum of a System

In terms of momentum of a system, internal forces, like all forces, always occur in action reaction pairs. Since the forces in action reaction pairs are equal and opposite due to Newton’s 3rd law, internal forces must always sum to zero. If the internal forces cancel, that means that the sum of the total forces must simply be all of the external forces. External forces could sum to zero but that won’t always be the case. 

The change in net momentum is directly related to the net external force. If the net external force acting on a system is equal to zero, then its net momentum is conserved so pinitial = pfinal. This only applies to the net momentum of a system, not individual momenta of each object. 

Example Problem #1:

A car of mass 1000 kg is traveling at a velocity of 50 m/s. The car has a passenger of mass 75 kg traveling with it. What is the total momentum of the car and the passenger?

Solution:

The momentum of the car is given by the formula: momentum = mass * velocity

The mass of the car is 1000 kg, and its velocity is 50 m/s.

Therefore, the momentum of the car is: momentum = 1000 kg * 50 m/s = 50000 kg*m/s

The momentum of the passenger is given by the formula: momentum = mass * velocity

The mass of the passenger is 75 kg, and its velocity is 50 m/s (since it is traveling with the car).

Therefore, the momentum of the passenger is: momentum = 75 kg * 50 m/s = 3750 kg*m/s

The total momentum of the car and the passenger is therefore: 50000 kgm/s + 3750 kgm/s = 53750 kg*m/s

Example Problem #2:

A spaceship of mass 1000 kg is traveling at a velocity of 50 m/s. The spaceship has a cargo of mass 500 kg traveling with it. The cargo has a velocity of 25 m/s relative to the spaceship. What is the total momentum of the spaceship and the cargo?

Solution:

The momentum of the spaceship is given by the formula: momentum = mass * velocity

The mass of the spaceship is 1000 kg, and its velocity is 50 m/s.

Therefore, the momentum of the spaceship is: momentum = 1000 kg * 50 m/s = 50000 kg*m/s

The momentum of the cargo is given by the formula: momentum = mass * velocity

The mass of the cargo is 500 kg, and its velocity relative to the spaceship is 25 m/s.

Therefore, the momentum of the cargo is: momentum = 500 kg * 25 m/s = 12500 kg*m/s

The total momentum of the spaceship and the cargo is therefore: 50000 kgm/s + 12500 kgm/s = 62500 kg*m/s

Example Problem #3:

A train of mass 10000 kg is traveling at a velocity of 50 m/s. The train has 20 cars, each with a mass of 1000 kg and a velocity of 50 m/s. What is the total momentum of the train and the cars?

Solution:

The momentum of the train is given by the formula: momentum = mass * velocity

The mass of the train is 10000 kg, and its velocity is 50 m/s.

Therefore, the momentum of the train is: momentum = 10000 kg * 50 m/s = 500000 kg*m/s

The total momentum of the cars is given by the formula: momentum = mass * velocity

The mass of each car is 1000 kg, and the velocity of each car is 50 m/s.

There are 20 cars, so the total mass of the cars is: mass = 20 cars * 1000 kg/car = 20000 kg

The total velocity of the cars is: velocity = 20 cars * 50 m/s/car = 1000 m/s

Therefore, the total momentum of the cars is: momentum = 20000 kg * 1000 m/s = 2000000 kg*m/s

The total momentum of the train and the cars is therefore: 500000 kg*m/s + 2000000 kg