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At a mail sorting facility, packages slide down a ramp but are stopped part way down the ramp so that they can be scanned. While the packages are scanned, they are held in place by a horizontal force from a spring‑loaded arm. Calculate the minimum force that must be applied by this arm to hold a package on the ramp. The ramp is at an angle of θ = 39.5 ∘ θ=39.5∘ with the horizontal. The package has a mass of 9.75 kg. 9.75 kg. The coefficient of static friction between the ramp and the package is 0.455 .

Answer :

davidlcastiblanco

Answer:

The minimum force mus be applied by this arm to hold is

Fa = 25.664 N

Explanation:

Given m = 9.75 kg  , µ = 0.455  ,θ = 39.5º

The force of weight can be model using  

Fw = m*g*sinθ

The force of the friction

Fr = µ*Fn

Fn = m*g*cosθ + Fa*sinθ

Replancing

Fr = µ*(m*g*cosθ + Fa*sinθ)

Fa*(cosθ + µ*sinθ) =  m*g*(sinθ - µ*cosθ)

Fa =  m*g*(sinθ - µ*cosθ)/(cosθ + µ*sinθ)

Replacing numeric to find Fa that is the minimum fort the must be applied  

Fa = [ 9.75 kg * 9.8 m/s² * (sin (39.5º) - 0.455 * cos (39.5º) ) ] / [ ( cos (39.5º) + 0.455 * sin (39.5º) ) ]

Fa = 25.664 N

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