The unstretched length of the spring is r. When pin P is in an arbitrary position theta, determine the x- and y-components of the force which the spring exerts on the pin. Evaluate your general expressions for r =400 mm, k = 1.4 kN/m, and theta = 40 degrees. (Note: The force in a spring is given by F= δk, where δ is the extension from the unstretched length.)
- NCSLv 72 months ago
force in the spring is
F = kδ
where δ is the elongation of the spring
for an angle Θ the vertical (down) and horizontal (x) displacement of the pin is
x = r*sinΘ
y = r*(1-cosΘ)
Then the length of the spring has y-component
Y = r + r*(1 - cosΘ) = r*(2 - cosΘ)
x = r*sinΘ
making the length of the spring
L = √(Y² + x²) = √[r²*(2-cosΘ)² + r²sin²Θ]
L = r*√(4 - 4cosΘ + cos²Θ + sin²Θ) = r*√(5 - 4cosΘ)
which in turn makes
δ = L - r = r*[√(5 - 4cosΘ) - 1]
(As a check, we note that for Θ = 0º we have
δ = r*[√(5-4) - 1] = r*0 = 0
So the magnitude of the force is
|F| = kδ = k*r*[√(5 - 4cosΘ) - 1]
The angle the spring makes above the -x axis is
φ = arctan(Y/x) = arctan(r(2 - cosΘ)/(r*sinΘ)
φ = arctan((2-cosΘ)/sinΘ)
so the x-component of the force is
Fx = k*r*[√(5 - 4cosΘ) - 1]*cosφ ← to the left
and the y-component is
Fy = k*r*[√(5 - 4cosΘ) - 1]*sinφ ← up
for φ = arctan((2-cosΘ)/sinΘ)
It doesn't look to me like this can be simplified.
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- Anonymous2 months ago
1) Not thermodynamics.
2) Diagram missing.