Physics tension problems can get extremely confusing if your method isn’t rock solid (especially when you’re a little sleep deprived).
And if you’ve ever been in a similar situation to this physicsforums commenter…
Then you already know what I’m talking about.
So below I’ll go through a solution to a tension/Newton’s Laws problem pulled from an actual Physics 1 exam.
And once you wrap your head around how to set up your equations so that you don’t get ass-backwards with the direction of your tension forces, you’ll never have an issue with them again!
Here’s what to do.
Let’s jump in…
Consider a force F pulling the rope as in the Figure. The rope is massless and the pulley is frictionless. The coefficient of friction between each box and the surface is uk = 0.05, mA = 2 kg and mB = 2 kg. Given that the two boxes move with an acceleration of magnitude a = 1 m/s^2, find the tensions T1 and T2 in the two ropes.
Following our problem solving process, first things first, let’s restate the problem and identify exactly what we’re trying to find.
We’re stating our assumptions (that the rope is massless and the pulley is frictionless), restating our known variables (the kinetic coefficient of friction, the two masses of the blocks, the acceleration of the system), and identifying the two unknowns that we need to find (the tension in each section of rope).
Then I’m taking a quick guess at what the two tensions are to establish a baseline we can compare the final answer to. I’m estimating that T2 will be approximately double T1 because it has to pull the mass of both the blocks, not just one.
Diagram & Variables
Even though the problem statement diagram is good, I’m re-creating it in my solution so that I can add in labels for each variable I think I need to consider in the problem. We know the coefficient of friction, the two masses, and the acceleration. And we’re also assuming that gravity acts downwards.
Additionally, we have the pulling force F that creates tension T1 in the rope between the two masses, and tension T2 in the rope that wraps around the pulley.
Free Body Diagram & Equations
Next, for any problem where you’re analyzing forces and using Newton’s laws to solve the problem, you should be constructing a set of free body diagrams so that you can visualize how, and in what direction, each force is acting on each mass in the system. This is what I’ve done below.
I’m then applying a set of force balance equations for each mass and using Newton’s 2nd Law to set up a system of equations that we can then use to solve for the two tension forces.
Keep in mind that because of Newton’s 1st Law, T1 acts to the right on mass A, while it acts to the left with an equal amount of force on mass B. This is because the rope itself is not accelerating in relation to the overall system (it remains taught as both masses continue to accelerate at 1 m/s^2), but is still providing the force required to keep mass A accelerating at 1 m/s^2.
Last, I’m applying the equation for kinetic friction to both of the masses, which acts in the opposite direction of the acceleration, and is proportional to the normal force acting on each mass (which ends up being equal to the weight because there are no other forces acting in the y-direction). This then gives us 6 equations with 6 unknowns, so we can then start to solve for T1 and T2.
Now that we have all of the equations that we need, I’m using algebra to re-arrange the mass A equations to form an equation for T1 in terms of our known variables (mA, a, uk, and g), and then the mass B equations to form an equation for T2 in terms of our known variables as well (mA, mB, a, uk, g).
Then it’s just a matter of plugging in our values and calculating out the numerical solutions for T1 and T2.
Comparing to my initial guess, it turns out that I was correct in assuming that T2 would be double T1, however my values were off. I underestimated the force and didn’t think to simply multiply the mass (2 kg) by the acceleration (1 m/s^2) and then add some extra force for friction.
Okay so thats it for this Physics 1 exam problem. All force/Newton’s Law problems (including tension problems) tend to have the same typical structure (diagram > FBD and force balance equations > re-arrange and solve) , and are some of the most popular exam questions out there in mechanics.
If you learn this general structure, and repeat the process with a number of different problems, you’ll be more than prepared when one of these pops up on your midterm or final.