- #1
songoku
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- 336
- Homework Statement
- (see picture)
Smaller block has mass 1 kg and larger block has mass 2 kg. Coefficient of friction between the two blocks is 0.2 and between larger block and floor is 0.3. Find the value of P so that the two blocks move relative to each other at constant sped
- Relevant Equations
- Newton's 2nd law of motion
Let:
Smaller block = m1 = 1 kg
Large block = m2 = 2 kg
Coefficient of friction between the two blocks = μ1 = 0.2
Coefficient of friction between larger block and floor = μ2 = 0.3
Tension connecting two blocks through two pulleys = T
Angle between tension and horizontal = θ = 37o
Friction between two blocks = f1
Friction between larger block and floor = f2
Normal force acting on m1 = N1
Normal force acting on m2 = N2
Weight of m1 = W1
Weight of m2 = W2
Free body diagram of m1:
T cos θ to the left
T sin θ upwards
W1 downwards
N1 upwards
f1 to the right
Equation of motion for m1:
T sin θ + N1 = W1
N1 = W1 - T sin θ ...(i)
f1 = μ1 . N1 = μ1 (W1 - T sin θ) ... (ii)
T cos θ = f1
T cos θ = μ1 (W1 - T sin θ)
T cos θ = μ1 . W1 - μ1 . T sin θ
T cos θ + μ1 . T sin θ = μ1 . W1
T = (μ1 . W1) / (cos θ + μ1 . sin θ) ... (iii)Free body diagram of m1:
f1 to the left
f2 to the left
P to the right
W2 downwards
N1 downwards
N2 upwards
T to the left
Equation of motion for m2:
$$P = f_1 + f_2 + T
\\= \mu_1 . W_1 - \mu_1 . T \sin \theta + \mu_2 . (W_2 + N_1) + \frac{\mu_1 . W_1}{\cos \theta + \mu_1 . \sin \theta}
\\= \mu_1 . W_1 - \mu_1 . \sin \theta . \frac{\mu_1 . W_1}{\cos \theta + \mu_1 . \sin \theta} + \mu_2 . \left( W_2 + W_1 - T \sin \theta \right) + \frac{\mu_1 . W_1}{\cos \theta + \mu_1 . \sin \theta}
\\=\mu_1 . W_1 - \mu_1 . \sin \theta . \frac{\mu_1 . W_1}{\cos \theta + \mu_1 . \sin \theta} + \mu_2 . \left( W_2 + W_1 - \frac{\mu_1 . W_1. \sin \theta}{\cos \theta + \mu_1 . \sin \theta} \right) + \frac{\mu_1 . W_1}{\cos \theta + \mu_1 . \sin \theta}
$$
Is this correct? When I plug the value I got 12 something Newton and the answer key is 15 NThanks