 #1
Ithilrandir
 82
 2
 Homework Statement:

A rocketpowered high speed sled running on rails employs pivot shoes between the sled and the rails, as shown in the figure. Each shoe has replaceable running pads at the heel and toe. The coefficient of friction between the running pads and the rail is μ. The rate at which the running pad material is worn off during the operation of the rocket sled is proportional to the friction force acting on the rubbing pad. If the shoe pivot point is at a given height h above the rail surface, at what horizontal distance, x, from the vertical centreline between the two rubbing pads should the pivot point P be placed do that the two rubbing pads will wear away at the same rate?
W: that portion of the weight of the rocket sled carried by the shoe in the diagram
H: horizontal component of force at the shoe pivot point
ℓ : total length between centres of the two rubbing pads
 Relevant Equations:
 ...
Letting the normal force on the pad on the left be 1 and the one on the right be 2,
Normal force R_{1} + R_{2} = W
Since the sled is not rotating, net moment =0,
Taking moment about the right pad,
W(ℓ/2 + x)  Hh  R_{1}ℓ =0
R_{1} = W/2 + Wx/ℓ  Hh/ℓ
Since the rate of the pads being worn is proportional to the frictional force, for the rate to be the same the frictional force on both sides needs to be the same as well.
R_{1} = R_{2} = W/2
Putting this back in the previous equation,
You get Wx/ℓ = Hh/ℓ,
x = Hh/W
However in the answer key the answer is μh. What am I missing in my steps?
Normal force R_{1} + R_{2} = W
Since the sled is not rotating, net moment =0,
Taking moment about the right pad,
W(ℓ/2 + x)  Hh  R_{1}ℓ =0
R_{1} = W/2 + Wx/ℓ  Hh/ℓ
Since the rate of the pads being worn is proportional to the frictional force, for the rate to be the same the frictional force on both sides needs to be the same as well.
R_{1} = R_{2} = W/2
Putting this back in the previous equation,
You get Wx/ℓ = Hh/ℓ,
x = Hh/W
However in the answer key the answer is μh. What am I missing in my steps?