the length of a rubber cord increases to 60 cm under a weight of 100g suspended from it and to 70cm under a weight of 120g.find the initial length of the cord .also calculate the weight under which the length of the cord will become 74 cm.
Answers
k = spring constant
L₀ = initial length of cord
L₁ = length of cord when 100 g weight is suspended = 60 cm
m = 100 g
using equilibrium of force
k (L₁ - L₀) = mg eq-1
L₂ = length of cord when 120 g weight is suspended = 70 cm
M = 120 g
using equilibrium of force
k (L₂ - L₀) = Mg eq-2
dividing eq-1 by eq-2
k (L₁ - L₀) /(k (L₂ - L₀)) = mg/(Mg)
(60 - L₀)/(70 - L₀) = 100/120
L₀ = 10 cm
using eq-1
k (L₁ - L₀) = mg
k (60 - 10) (0.01) = (0.1) (9.8)
k = 1.96 N/m
let the weight hanged be "m' "
L₂ = 74 cm
using equilibrium of force
k (L₂ - L₀) = m' g
(1.96) (0.74 - 0.1) = m' (9.8)
m' = 0.128 kg
m' = 0.128 x 1000 g
m' = 128 g
Explanation:
Let the initial length of the rubber cord be x cm. Let k be the spring constant of the rubber cord.
When a weight of 100g is suspended from the cord, the extension is (60 - x) cm. Using Hooke's law, we have:
F = k * (60 - x) 100g = 0.1kg * 9.8m/s^2 = 0.98N 0.98N = k * (60 - x)
Similarly, when a weight of 120g is suspended from the cord, the extension is (70 - x) cm. We have:
F = k * (70 - x) 120g = 0.12kg * 9.8m/s^2 = 1.176N 1.176N = k * (70 - x)
Solving these two equations, we get:
k = 0.98N / (60 - x) = 1.176N / (70 - x)
Simplifying, we get:
5880 - 98x = 7056 - 84x 14x = 1176 x = 84 cm
Therefore, the initial length of the rubber cord is 84 cm.
To find the weight under which the length of the cord will become 74 cm, we can use the equation:
F = k * (74 - x)
Substituting the value of k, we get:
F = 0.98N / (60 - x) * (74 - x)
Simplifying, we get:
F = 2.058N
Therefore, the weight under which the length of the cord will become 74 cm is:
W = F / g = 2.058N / 9.8m/s^2 = 0.210 kg = 210g