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PHYS101 Problem with static frictional force
Problem with static frictional force
The figure below shows a block of mass m connected to a block of mass M = 2.00kg, both on45◦ inclined planes where the coefficient of static friction is 0.28. Find the minimum and themaximum values for m for which the system is at rest.
45° 45°
m M
The masses m and M are on the verge to slide up or down for the minimum and the maximummass of m. If we want to determine the minimum mass, obviously the mass M will be on theverge to slide down and the mass of m will be on the verge to slide up. Therefore the staticfrictional force for M will be parallel to the tension of the cord, whereas the the static frictionalforce for m will be antiparallel to the tension of the cord and also antiparallel to the potentialdirection of the motion.Let us now consider the Free Body Diagrams for m and M in the case of the minimum massof m.
From the Free Body Diagram for m weget:
∑ Fy = Fn −mg cos θ = 0
⇒ FN = mg cos θ
∑ Fx = T −mg sin θ − fs = 0
with fs = µsFN we get:
T −mg sin θ − µsmg cos θ = 0 (1)
From the Free Body Diagram for M weget:
∑ Fy = Fn −Mg cos θ = 0
⇒ FN = Mg cos θ
∑ Fx = −T −Mg sin θ − Fs = 0
with Fs = µsFN we get:
−T + Mg sin θ − µsmg cos θ = 0 (2)
c©2014 Department of Physics, Eastern Mediterranean University Page 1 of 2
PHYS101 Problem with static frictional force
(1) +(2) gives−mg(sin θ + µs cos θ) + Mg(sin θ − µs cos θ) = 0
Solving this equation for m gives:
m =sin θ − µs cos θ
sin θ + µs cos θM =
sin 45◦ − 0.28 cos 45◦
sin 45◦ + 0.28 cos 45◦2.00kg = 1.125kg
If we want to determine the maximum mass, obviously the mass M will be on the verge toslide up and the mass of m will be on the verge to slide down. Therefore the static frictionalforce for M will be antiparallel to the tension of the cord, whereas the the static frictional forcefor m will be parallel to the tension of the cord and also parallel to the potential direction ofthe motion.Let us now consider the Free Body Diagrams for m and M in the case of the maximum massof m.
From the Free Body Diagram for m weget:
∑ Fy = Fn −mg cos θ = 0
⇒ FN = mg cos θ
∑ Fx = T −mg sin θ+ fs = 0
with fs = µsFN we get:
T −mg sin θ+µsmg cos θ = 0 (3)
From the Free Body Diagram for M weget:
∑ Fy = Fn −Mg cos θ = 0
⇒ FN = Mg cos θ
∑ Fx = −T −Mg sin θ+Fs = 0
with Fs = µsFN we get:
−T + Mg sin θ+µsmg cos θ = 0 (4)(1) +(4) gives
−mg(sin θ−µs cos θ) + Mg(sin θ+µs cos θ) = 0
Solving this equation for m gives:
m =sin θ+µs cos θ
sin θ−µs cos θM =
sin 45◦+0.28 cos 45◦
sin 45◦−0.28 cos 45◦2.00kg = 3.28kg
c©2014 Department of Physics, Eastern Mediterranean University Page 2 of 2
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