Math Problem Statement

A simple harmonic oscillator moves under the influence of Hooke's law with force F with arrows = −kx with arrow where x with arrow is the displacement from equilibrium. The spring constant k is given by the following equation. k =

F with arrows

x with arrow

= Fs x When an object of mass m = 9 g hangs from the spring, the force stretching the spring is Fs = mg =

⨯ 10−3 kg

9.80 m/s2

= N.

Solution

Let’s solve this step by step.

  1. Spring Force Formula: According to Hooke's Law, the force stretching the spring is proportional to the displacement and can be expressed as Fs=kxF_s = kx. The spring constant kk can then be derived as k=Fsxk = \frac{F_s}{x}.

  2. Weight of the Mass: The force FsF_s stretching the spring due to gravity is given by: Fs=mgF_s = mg where mm is the mass of the object and gg is the acceleration due to gravity.

  3. Substitute Values for FsF_s: The mass m=9g=9×103kgm = 9 \, \text{g} = 9 \times 10^{-3} \, \text{kg}, and g=9.80m/s2g = 9.80 \, \text{m/s}^2.

    Calculate FsF_s: Fs=(9×103)9.80=0.0882N.F_s = (9 \times 10^{-3}) \cdot 9.80 = 0.0882 \, \text{N}.

Final Answer:

The force stretching the spring is: Fs=0.0882N.F_s = 0.0882 \, \text{N}.

Let me know if you would like assistance in calculating the spring constant kk or solving related problems.

Follow-Up Questions:

  1. How is the displacement xx related to the spring constant kk?
  2. What would be the spring constant if the displacement xx is given as 0.01m0.01 \, \text{m}?
  3. How does increasing the mass affect the stretching force on the spring?
  4. Can you derive the formula for kk in terms of mass and displacement?
  5. What assumptions are made in this problem for simplicity?

Tip: Always keep units consistent while solving physics problems!

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Math Problem Analysis

Mathematical Concepts

Classical Mechanics
Hooke's Law
Force Calculation
Unit Conversion

Formulas

Hooke's Law: F = kx
Spring Constant: k = F/x
Force due to Gravity: F = mg

Theorems

Hooke's Law

Suitable Grade Level

Grades 9-12