PhysicsLAB: Systems of Bodies

When two or more bodies are connected by a cord and move in tandem, then they are considered to be a system of bodies. When working problems involving systems of bodies,

the first step is to draw freebody diagrams for each object. Remember, that the principle forces that we are now considering are: normals, weight, friction, tensions and generic applied forces.
since both masses are attached, they share the same kinematics properties: displacement, velocity, acceleration and time.
when writing equations for net F = ma, the direction of motion is considered to be the positive direction for the acceleration of the objects comprising the system.

Table Surface

Let's consider as our first example, two identical objects being dragged across the surface of a frictionless table by two cords.

The two freebody diagrams would look like:

left mass		right mass

In order to determine the acceleration of the system and the tension in each cord, we will need to write the equations of motion for each mass: net F_x = ma_x and net F_y = ma_y.

	left mass	right mass
net F_x = ma_x	T₁ = ma	T₂ - T₁ = ma
net F_y = ma_y	- mg = 0	- mg = 0

Solving the two equations for net F_x = ma_x simultaneously yields the equation:

T₁ = ma

T₂ - T₁ = ma

-------------------

T₂ = 2ma

If the numerical values for T₂ and m are given, then this equation will allow you to solve for the acceleration of the system. Once the acceleration is known, then the tension in cord #1 can also be calculated.

Hanging Masses

Now suppose that the right mass is hanging off a frictionless table and the cord connecting it to the left mass passes over a "massless, frictionless" pulley.

How would this change the freebody diagrams for each mass and their equations of motion?

	mass on table	hanging mass

net F_x = ma_x	T = ma	------
net F_y = ma_y	- mg = 0	mg - T = ma

Solving these equations simultaneously for the acceleration yields the equation:

T = ma

mg - T = ma

--------------------

mg = 2ma

a = ½g

Once again, if the numerical value for m is given then this equation will allow you to solve for the acceleration of the system. Remember that "g" represents the acceleration due to gravity, 9.8 m/sec². Once the acceleration is known, then the tension in the cord can be calculated.

Refer to the following information for the next six questions.

Obviously, the masses in the previous example do not have to be the same. Solve for the acceleration of the system and the tension in the cord if the hanging object has a mass of 3 kg and the mass on the table is 2 kg.

	Write the equation for net F_x = ma_x for the 2 kg mass.

	Write the equation for net F_y = ma_y for the 2 kg mass.

	Write the equation for net F_x = ma_x for the 3 kg mass.

	Write the equation for net F_y = ma_y for the 3 kg mass.

	What is the acceleration of the system?

	What is the tension in the cord?

Atwood Machine

Refer to the following information for the next six questions.

In our third example the two masses are attached to the ends of a single cord that passes over a massless, frictionless pulley suspended from the ceiling. This situation is called an Atwood Machine.

	Write the equation for net F_y = ma_y for the 2 kg mass.

	Write the equation for net F_y = ma_y for the 5 kg mass.

	Why did we not need to subscript the tension variables?

	Why did we not need to write the equations for net F_x = ma_x for these two bodies?

	What is the acceleration of the system?

	What is the tension in the cord?