Connected Objects

When two objects are connected by a rope or in direct contact, they often move together with the same acceleration. This shared motion is the key constraint.

Draw separate free body diagrams (FBDs) for each object. Each object has its own forces—gravity, normal forces, tension, friction, etc.—but the connection ensures they accelerate together. For objects connected by a rope or moving together, the accelerations are equal: $a_1 = a_2 = a$.

Example: Two masses $m_1$ and $m_2$ are connected by a rope. A force $F$ pulls mass $m_2$, so both masses accelerate together with the same $a$.

Mass $m_1$ (on the left) experiences only tension $T$ from the rope pulling it forward. Mass $m_2$ (on the right) experiences the applied force $F$ pulling it forward and tension $T$ from the rope pulling it backward. The constraint $a_1 = a_2$ connects the two diagrams and lets you solve for tension and acceleration.

Let's examine both masses separately first:

Mass 1:

Mass 2:

System vs Individual Approach:

You can analyze each mass separately or treat the connected objects as a single system. When analyzing separately, you apply Newton's second law to each object: $F_{\text{net},1} = m_1 a$ and $F_{\text{net},2} = m_2 a$. These equations involve the internal force (tension $T$ or contact force).

Treating them as a system eliminates internal forces like $T$ because they are equal-and-opposite pairs (Newton's third law). Applying Newton's second law to the whole system gives:

$F_{\text{ext}} = (m_1 + m_2)a$.

This directly gives acceleration $a$.

FBD for Combined System: