Pulley, Table & Friction - Dynamics

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Working through Newton's second law for a hanging mass connected by a rope and pulley to a box on a table — first with friction, then on an inclined ramp.

Box on Table — Has Friction

coefficient of friction = μ_k

Free body diagram

F_net = ma

make clockwise +

Mg − μ_kmg = (m + M)a

a = g(M − μ_km) ⁄ (M + m)

Tension force in rope?

↓

Rule — One rope, one tension

make clockwise +

F_net = ma

T − μ_kmg = ma

T = ma + μmg

Ex) T in Terms of M

Draw free body diagram for M

↓

Free body diagram of hanging mass M with tension T and weight Mg

ΣF_y = Ma

Mg − T_f = Ma

T_f = Mg − Ma

Massless, Frictionless Pulley and Ramp

Find the acceleration of the red box up the ramp (M)

Hanging mass m connected over a pulley to box M on a frictionless inclined ramp

Masses move together with one acceleration.

make CCW +

F_net = ma

mg − Mgsinθ = (m + M)a

Hanging weight force friction = ma

a = (m − Msinθ)g ⁄ (M + m)

When does a = 0?

↓

Msinθ = m

When does the box move down the ramp?

↓

a = (m − Msinθ)g ⁄ (M + m)

When mg greater than Msinθ > m