Force to make 2 blocks move relative to each other at constant speed

[songoku]

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 = 37^o
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

 

[haruspex]

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

View attachment 263108
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 = 37^o
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

Your working looks correct, and produces the right answer for theta=0, and also for W1=0.
I agree with your 12.something, depending on the value you use for g.

 

[songoku]

Thank you very much haruspex