Projectile motion of a two-point rigid body

[luinthoron]

I would like to patch some gaps in my physics background. For example, I've been trying to come up with the sollution to the following: I have a model rigid body made up of two mass points and a massless rod connecting them. I throw the body with initial velocity under some angle of elevation. My problem is the transfer from single mass point physics to rigid body physics. How do I get the actual movement of the rigid body from knowing the outer forces acting on the two masses? I guess I will have to work with the motion of the center of mass and rotation of the object separately, right?

Could you please at least show me the right direction of thought? Thanks a bunch.

 

[PeroK]

How do I get the actual movement of the rigid body from knowing the outer forces acting on the two masses? I guess I will have to work with the motion of the center of mass and rotation of the object separately, right?

Yes. Gravity acts on the centre of mass, so a body under gravity will translate but not rotate. If you drop a horizontal bar with a large mass at one end and a smaller mass at the other, then (ignoring air resistance) the bar remains horizontal as it falls.

 

[jbriggs444]

Modelling uniform gravity? Gravity plus air resistance? Non-uniform gravity? Modeling additional ad hoc forces and torques not necessarily confined to the plane of rotation?

 

[luinthoron]

Modelling uniform gravity? Gravity plus air resistance? Non-uniform gravity? Modeling additional ad hoc forces and torques not necessarily confined to the plane of rotation?

Uniform gravity plus air-resistance confined to the plane of rotation, for start. But ultimately, I would like to be able to solve the rigid body motion under any kind of given forces.

 

[luinthoron]

Yes. Gravity acts on the centre of mass, so a body under gravity will translate but not rotate. If you drop a horizontal bar with a large mass at one end and a smaller mass at the other, then (ignoring air resistance) the bar remains horizontal as it falls.

Fair point, but if I add air-resistance?

 

[jbriggs444]

Fair point, but if I add air-resistance?

How will you model the air resistance? Linear? Quadratic? Combined?

You are quickly approaching the point where a differential equation may not have a closed form solution.

 

[luinthoron]

How will you model the air resistance? Linear? Quadratic? Combined?

You are quickly approaching the point where a differential equation may not have a closed form solution.

I see. Since my goal is to practice the method on a simple situation first, I would go with linear air resistance first, regardless of the physical reality. I should also add that I am working with simple numerical solutions, first-order or second-order numerical methods. I could go further in precision if needed. My animations are for high-school class demonstration, so they do not have to be exact, but show the correct qualitative behaviour.

 

[jbriggs444]

I see. Since my goal is to practice the method on a simple situation first, I would go with linear air resistance first, regardless of the physical reality. I should also add that I am working with simple numerical solutions, first-order or second-order numerical methods. I could go further in precision if needed. My animations are for high-school class demonstration, so they do not have to be exact, but show the correct qualitative behaviour.

Given linear air resistance, one should be able to separate the calculation for rotation from the calculation for the trajectory of the center of mass. This follows since the added air resistance due to the rotation of one end is equal and opposite to the added air resistance of the other end.

You should be able to solve and see that the rotation will follow a pattern of exponential decay. You should be able to see the same for the linear motion, except that it will converge on a fall at terminal velocity.

In addition to rotational and linear motion being separable, linear drag should allow you to treat movement in the x and y directions independently.

I am very rusty at this stuff (unpracticed for 40 years or so) but may be able to guide you through writing and solving the differential equations of motion. Certainly there are others around here who can.

 

[anorlunda]

My animations are for high-school class demonstration, so they do not have to be exact, but show the correct qualitative behaviour.

Then I would recommend keeping it simpler, and not trying to consider all forces at once.

One simple case is a body in free fall in a vacuum. Only the center of gravity is important.

Another simple case is a body subjected to a single force that is not through the center of gravity. It will start rotating, and maybe moving also, depending on the angle of the force.

Then tell students that for linear systems, solutions can be superimposed. That brings two more important concepts in: linear and superposition.