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A 50 N Collar Slides Without Friction Along a Smooth and Which is Kept Inclined at 60o to the Horizontal.The Spring Attached to the Collar and the Support C.The Spring is - Engineering Mechanics

Answer in Brief

A 50 N collar slides without friction along a smooth and which is kept inclined at 60o to the horizontal.
The spring attached to the collar and the support C.The spring is unstretched when the roller is at A(AC is horizontal).
Determine the value of spring constant k given that the collar has a velocity of 2.5 m/s when it has moved 0.5 m along the rod as shown in the figure.

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Solution

Given : W=50 N
AB = AC = 0.5 m
To find : Spring constant
Solution :
Mass of collar = `50/g` kg
Let us assume that h = 0 at position 2
POSITION 1 :
x = 0
Es1 = `1/2` x k x x12 = 0
h1 = 0.5sin60 = 0.433 m
PE1=mgh1=21.65 J
vA = 0 m/s
KE1 = 0J
POSITION II :
vB = 2.5 m/s
PE2 = mgh = 0 J (because h=0)
KE2 = `1/2` 𝑋 π‘š 𝑣2 = 12 𝑋 `50/g` 𝑋 2.52
= 15.9276 J
In β–³ABC
Applying cosine rule
BC2 = AB2 + AC2 -2 X AB X AC X cos(BAC)
= 0.52 + 0.52 – 2 x 0.5 x 0.5 x cos120
= 0.75
BC = 0.866 m
Un-stretched length of the spring = 0.5 m
Extension of spring(x) = 0.866 - 0.5
=0.366 m
Es2 = `1/2` x k x22
= 0.067k
APPLYING WORK ENERGY PRINCIPLE
U1-2 = KE2 - KE1
PE1 - PE2 + ES1 - ES2 = KE2 - KE1

21.6506-0+0-0.067K=15.9276-0

K = 85.4343 N/m
SPRING CONSTANT IS 85.4343 N/m

Concept: Principle of Work and Energy
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