Impulse-Momentum for a Particle (Direct application of Euler's 1st Law)
Example: A time-dependent force, F(t), acts on block A shown in the figure below is initially
at rest on the incline plane where θ = 30o . The mass of the block m = 1 slug. F(t) = 30 + t lb.
The coefficient of friction μ is 1/(2√3). Apply Euler’s 1st Law directly to find the acceleration of
block A up the plane if, indeed, it does occur. If so, then determine the speed of the block after
2 seconds.
Strategy: First determine if the force is sufficient to propel the block up the incline by checking
for motion starting with a free body diagram of block A and by finding the acceleration of block A
at t = 0. Then, if aA > 0, integrate to find the speed of block A after 2 seconds.
Σ Fx = m aA 30 + t ˗ W sinθ ˗ f = m aA 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 aA = m dv/dt
And for the data 5.85 + t = aA = dv/dt So aA > 0 at t = 0.
Integrate 30 + t ˗ W sinθ ˗ μ W cosθ = m aA to obtain
2
∫ [ 30 + t ˗ W sinθ ˗ μ W cosθ ] dt = m v2 ˗ mv1
0
Input the data W = 32.2 lb, θ = 30o and μ = 1/(2√3) which gives
∫ (5.85 + t) dt = m v2 – mv1 = v2 ˗ 0
Integration gives 5.85 t + t2/2 | = 13.7 = v2 (same result)
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Copyright © 2019 Richard C. Coddington All rights reserved.
