Free-body diagram with pulley

Question

An object with a mass M = 250 g is at rest on a plane that makes an

angle θ = 30º above the horizontal. The coefficient of kinetic friction between M and the plane is µk = 0.100. Mass M is attached by a string to another mass, m = 300 g, which hangs freely. When mass m has fallen 30.0 cm, its speed is?

 

I'm told the answer is 128cm/s, but I keep getting 12.8 cm/s. I don't know where my mistake is?

 

Thank you in advance.

Answer 1

M = 0.250 kg

m = 0.300 kg

θ = 30° 

μ_k = 0.100

Δy = -30 cm = -0.30 m

 

First, find the acceleration of m, and then use a kinematic relation to find its final speed.

 

m has tension T pulling up, opposed by -mg pulling down.  The acceleration is -a (vertically down).

 

T - mg = -ma ⇒ T - 0.300(9.8) = -0.300a ⇒

 

(i)  T - 2.94 = -0.3a

 

M has the same tension T pulling up the plane, opposed by -Mgsinθ and -μ_kMgcosθ pointing down the plane.  Its acceleration is +a (up the plane).

 

T - Mgsinθ - μ_kMgcosθ = Ma ⇒

 

T - 0.25(9.8)(0.5) - 0.1(0.25)(9.8)(0.866) = 0.25a ⇒

 

(ii)  T - 1.437 = 0.25a

 

Subtract (ii) from (i) to eliminate T.

 

-1.503 = -0.55a

 

a = 1.503/0.55 m/s² = 2.733 m/s²

 

Kinematic relation for m:

 

v² - 0² = 2(-a)Δy = 2(-2.733)(-0.30) (m/s)²

 

v = √1.6398 m/s ≅ 1.28 m/s = 128 cm/s