Two masses are connected by a string which goes over an ideal pulley as shown. Block A has a mass of 3.00 kg and can slide along a rough plane inclined 30.0° to the horizontal. The coefficient of kinetic friction between block A and the plane is 0.4. Block B has a mass of 2.77 kg. What is the acceleration of the blocks?

0.392 m/s²

Explanation:

Draw a free body diagram for each block.

Block A has 4 forces acting on it:

Weight force Mg pulling straight down,

Normal force N pushing perpendicular to the incline,

Friction force Nμ pushing parallel up the incline,

Tension force T pulling parallel up the incline.

Block B has 2 forces acting on it:

Weight force mg pulling straight down,

Tension force T pulling straight up.

Sum of forces on block B in the -y direction:

∑F = ma

mg − T = ma

T = mg − ma

Sum of forces on block A in the perpendicular direction:

∑F = ma

N − Mg cos 30° = 0

N = Mg cos 30°

Sum of forces on block A in the parallel direction:

∑F = ma

T − Nμ − Mg sin 30° = Ma

Substitute:

mg − ma − Mgμ cos 30° − Mg sin 30° = Ma

g (m − Mμ cos 30° − M sin 30°) = (m + M) a

a = g (m − Mμ cos 30° − M sin 30°) / (m + M)

Plug in values:

a = (9.8) (2.77 − (3.00)(0.4)(cos 30°) − (3.00)(sin 30°)) / (2.77 + 3.00)

a = 0.392 m/s²

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