How Free-Body Diagrams Determine Tension in Connected Systems

Tension comes from the connected system

Tension is not automatically equal to an object's weight. It is an interaction force set by the motion of the entire connected system. The safest method is to draw a separate free-body diagram for each object and apply Newton's second law along the direction of the string.

Four steps that work across pulley problems

1. Choose a direction for the shared acceleration. If the result is negative, the real direction is opposite your choice.
2. Isolate each object. Include only forces acting on that object.
3. Write one force equation per object. Keep the sign convention consistent.
4. Use the string constraint. A taut, massless, inextensible string over an ideal pulley gives connected objects the same acceleration magnitude and the same tension along the string.

Fresh example: one block on a table

A $3.0\,\mathrm{kg}$ block sits on a frictionless horizontal table and is connected over an ideal pulley to a hanging $2.0\,\mathrm{kg}$ block. Let the table block accelerate right and the hanging block accelerate down.

For the table block,

$$T=m_1a.$$

For the hanging block,

$$m_2g-T=m_2a.$$

Adding the equations gives

$$a=\frac{m_2g}{m_1+m_2} =\frac{(2.0)(9.8)}{3.0+2.0} =3.92\,\mathrm{m/s^2}.$$

Then

$$T=m_1a=(3.0)(3.92)=11.8\,\mathrm{N}.$$

Check the result from the hanging mass: $T=m_2(g-a)=11.8\,\mathrm{N}$. This second calculation catches many sign errors.

When one tension is not enough

The equal-tension shortcut depends on the ideal model. A pulley with rotational inertia generally needs different tensions on its two sides to create angular acceleration. A massive or stretching string can also make tension vary with position. State the model before using a single $T$ everywhere.