# Direction of tension with spring force and torque involved

Can anyone please help explain why the arrows in this diagram are drawn like they are? I'm having a tough time understanding why they are like this. I really want to understand this before I tackle the equations. Also looking at the diagram would the tension (T1 and T2) just be spring force? Can someone please help, I'm pretty confused over this.

You do not have to worry about the direction the tensions point to. Just write your equations and if one of them is pointing in a wrong direction then it will come up with a minus.

• Thanks for the answer, but how will it come up with a minus, for example the answer for one of the equations is T1=K2(rθm-y), but how do I know this to be true just by looking at the arrows? Jan 23, 2022 at 21:26
• Also is T1 and T2 just there to represent the spring force? Jan 23, 2022 at 21:30
• In this sort of problem you will end up with a system of n linear equations with n unknowns. Some of your unknowns may come out negative after you solve your system. For the negative ones, just switch the signs. Now tensions actually always pull. You cannot push a rope. So the correct way to think about it is that the spring is pulled to the left by T1 and to the right also by T1. Same with the mass M is pulled to the left by T2 and pulled to the right by T1. At the object, the "ropes" are always pulling away, so you may even think of the arrows pointing both ways, depending where they act on. Jan 23, 2022 at 21:31

The tension in a string has no inherent direction. It's just the force with which the lengths of string to the right and left of a point are trying to pull the string apart. When we draw arrows to indicate tension it should be done in such a way that what force they are representing is clear. It is not clear in this figure.

The opposite arrows for $$T_1$$ on the top line represent the forces the string exerts on the pulleys at the ends. The bottom line is more problematic. If the spring $$K_1$$ is massles the tension will be the same to right and left of the spring, but the middle $$T_2$$ should point the other way so as to represent the force on the block. The resulting equation of motion should be $$M\ddot y= T_1-T_2$$

The string/rope transmits forces between the components as shown below.

The string/rope and the springs are assumed to be massless.