Solve question 5on elasticity

Economics

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2 parts
a

50 N\text{50 N}

b

2.0 m\text{2.0 m}

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Here is the solution to Question 5 on elasticity:

Part 5

A light elastic string has natural length 0.8 m and modulus of elasticity 100 N.

Part a

50 N\text{50 N}

Find the tension in the string when its length is 1.2 m.

  1. Identify the given values.

    Natural length, L0=0.8L_0 = 0.8 m
    Modulus of elasticity, λ=100\lambda = 100 N
    Extended length, L=1.2L = 1.2 m

  2. Calculate the extension (ee).

    The extension is the difference between the extended length and the natural length.

    e=L−L0e = L - L_0 e=1.2 m−0.8 me = 1.2 \text{ m} - 0.8 \text{ m} e=0.4 me = 0.4 \text{ m}
  3. Apply Hooke's Law to find the tension (TT).

    Hooke's Law states that T=λeL0T = \frac{\lambda e}{L_0}.

    T=(100 N)(0.4 m)0.8 mT = \frac{(100 \text{ N})(0.4 \text{ m})}{0.8 \text{ m}} T=400.8 NT = \frac{40}{0.8} \text{ N} T=50 NT = 50 \text{ N}

    The tension in the string is 50 N\boxed{\text{50 N}}.

Part b

2.0 m\text{2.0 m}

Find the length of the string when the tension is 150 N.

  1. Identify the given values.

    Natural length, L0=0.8L_0 = 0.8 m
    Modulus of elasticity, λ=100\lambda = 100 N
    Tension, T=150T = 150 N

  2. Apply Hooke's Law to find the extension (ee).

    Rearrange Hooke's Law (T=λeL0T = \frac{\lambda e}{L_0}) to solve for ee:

    e=TL0λe = \frac{T L_0}{\lambda} e=(150 N)(0.8 m)100 Ne = \frac{(150 \text{ N})(0.8 \text{ m})}{100 \text{ N}} e=120100 me = \frac{120}{100} \text{ m} e=1.2 me = 1.2 \text{ m}
  3. Calculate the extended length (LL).

    The extended length is the natural length plus the extension.

    L=L0+eL = L_0 + e L=0.8 m+1.2 mL = 0.8 \text{ m} + 1.2 \text{ m} L=2.0 mL = 2.0 \text{ m}

    The length of the string when the tension is 150 N is 2.0 m\boxed{\text{2.0 m}}.

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