Determine acceleration of weight, Mechanical Engineering

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Determine acceleration of weight:

Two pulleys of radii 200 mm and 400 mm are mounted co-axially and rigidly on a common-shaft.

These pulleys have a overall mass of 28 kg and radius of gyration of 200 mm of the combination. If weights W1 = 40 N and W2 = 50 N are suspended from points A1 and A2 by ropes which are wound round the two pulleys, determine acceleration of weight W2. Suppose g = 10 m/sec2.

Solution

 Clockwise moment of W1 about centre of pulley

= 40 × 0.4 = 16 N.m.

Anticlockwise moment of W2 about centre of pulley = 10 N.m.

Therefore the pulley shall rotate clockwise with angular acceleration α about the centre. After some time, W2 shall have moved a distance s2 with acceleration a2 and shall have velocity of v2  = r2  ω , while W1 will move

Distance  s1 = 2 s2  = 2 r2  θ .

with final velocity V1 = r1  ω = 2 V2

with acceleration .a1 = r1  α = 2 a2

Since,   rk  = 0.2 m.

I (m) = 28 × (0.2)2       = 1.12 kg m2

ω2 = 2 × α θ

By Using Principle of Conservation of Energy, and letting datum level as (A - A),

1561_Determine acceleration of weight.png

40 (s1 ) + 50 (so) = 50 (s1/2)  + 50 (so) + (40/10) × (V1)2 /2+ 50 × (V2)2 /2+( Im ω2 )/2

40 (s ) - 25 s = ω2 [2  × (0.4)2 + 2.5 × (0.2 )2 +(1.12/2)]

15 × (0.4 θ) = 2 α θ [0.32 + 0.1 + 0.56]

∴ α = 3.06 rad / sec 2

α ≈ 3 rad / sec 2 .


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