Impulse-Momentum for a Particle

Example: A time-dependent force, F(t), acts on block A shown in the figure below. The mass of

the block m = 1 slug. F(t) = 30 + t lb. The coefficient of friction μ is 1/(2√3). First show that the force is sufficient to propel the block up the incline starting at t = 0 where θ = 30^o. Then using linear impulse and momentum determine the speed of block A after 2 seconds.

Strategy: Check for motion starting up the plane with a free body diagram.

Σ F_x = m a_A 30 + t ˗ W sinθ ˗ f = m a_A For motion up the plane the force must

be sufficient to overcome the maximum friction force and the component of weight down

the plane. So f = μN = μ W cosθ.

30 + t ˗ W sinθ ˗ μ W cosθ = m a_A

Input the data W = 32.2 lb, θ = 30^o and μ = 1/(2√3) which gives 5.85 + t = a_A

So at t = 0 a_A > 0 and the block starts to slide up the plane. (result)

The linear impulse equals the change in linear momentum.

2

∫ (5.85 + t) dt = m v₂ – mv₁ = v₂

0

2

Integration gives 5.85 t + t²/2 | = 13.7 = v₂ (result)

0

Click here to solve this example using Euler’s 1^st Law directly.

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