Pulley & Table - Dynamics | Show Me The Physics

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Two bodies connected by one string move as one mass, with one shared acceleration — a block on a table, a mass hanging over a pulley, and one rope tying their motion together.

Find a for This System

Play with the simulation, then work through the derivation below

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Free body diagram of a block on a table connected over a pulley to a hanging mass

Two bodies move as one mass w/ one accel.

Newton's Second Law for the System

Free body diagram of the pulley-table system

ΣF_net = ma

Let's make clockwise +

Mg = (m + M)a

The 2 masses get their accel. from hanging mass.

Applies only for frictionless surface.

a = Mg / (M + m)

What If the Roles Were Reversed?

What would the accel. be if the little mass was hanging?

↓

a = mg / (M + m)

What Is the Tension Force of the Rope?

What is the tension force of rope?

↓

ΣF_x = ma

T_f = ma

Rule — One Rope, One Tension

Free body diagram used to find the rope tension

Let's make clockwise +

Tension in Terms of M

ΣF_y = Ma

Mg − T_f = Ma

Mg − Ma = T_f

T_f = M(g − a)

If Box on Table Had Friction

Free body diagram of the block on the table

coefficient of friction = μ

Free body diagram of the block on the table with friction