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Thursday, May 16, 2024

five problems designed to test students' ability to use fundamental physics equations with the requirement of algebraic transformations

 Here are five problems designed to test students' ability to use fundamental physics equations with the requirement of algebraic transformations:

  1. Problem: Projectile Motion A ball is launched from the ground at an angle 𝜃 with an initial velocity 𝑣0. It reaches a maximum height 𝐻.

    • Given: 𝐻 and 𝜃
    • Find: The initial velocity 𝑣0
    • Formula Involved: 𝐻=𝑣02sin2𝜃2𝑔

    Solution Steps:

    • Start with the given formula: 𝐻=𝑣02sin2𝜃2𝑔
    • Rearrange to solve for 𝑣0:
      𝑣02=2𝑔𝐻sin2𝜃
      𝑣0=2𝑔𝐻sin2𝜃
  2. Problem: Kinetic and Potential Energy Conversion A block of mass 𝑚 slides down a frictionless incline of height starting from rest. Find the speed of the block at the bottom.

    • Given: 𝑚,
    • Find: The speed 𝑣 at the bottom
    • Formula Involved: 𝑚𝑔=12𝑚𝑣2

    Solution Steps:

    • Start with the energy conservation equation: 𝑚𝑔=12𝑚𝑣2
    • Cancel out 𝑚 from both sides:
      𝑔=12𝑣2
      𝑣2=2𝑔
      𝑣=2𝑔
  3. Problem: Uniform Circular Motion An object is moving in a circle of radius 𝑟 with a constant speed 𝑣. Find the centripetal force 𝐹𝑐 acting on the object.

    • Given: 𝑚, 𝑣, 𝑟
    • Find: The centripetal force 𝐹𝑐
    • Formula Involved: 𝐹𝑐=𝑚𝑣2𝑟

    Solution Steps:

    • Use the centripetal force formula: 𝐹𝑐=𝑚𝑣2𝑟
    • This problem is straightforward and requires just one algebraic step, but students can be asked to first derive 𝑣2 if given 𝑇 (the period):
      𝑣=2𝜋𝑟𝑇
      𝑣2=(2𝜋𝑟𝑇)2=4𝜋2𝑟2𝑇2
      𝐹𝑐=𝑚4𝜋2𝑟2𝑟𝑇2=4𝜋2𝑚𝑟𝑇2
  4. Problem: Newton’s Second Law with Friction A block of mass 𝑚 is pushed across a horizontal surface with a constant force 𝐹. The coefficient of kinetic friction between the block and the surface is 𝜇𝑘. Find the acceleration 𝑎 of the block.

    • Given: 𝐹, 𝑚, 𝜇𝑘
    • Find: The acceleration 𝑎
    • Formula Involved: 𝐹𝐹friction=𝑚𝑎 and 𝐹friction=𝜇𝑘𝑚𝑔

    Solution Steps:

    • Start with the force balance: 𝐹𝜇𝑘𝑚𝑔=𝑚𝑎
    • Rearrange to solve for 𝑎:
      𝑎=𝐹𝜇𝑘𝑚𝑔𝑚
      𝑎=𝐹𝑚𝜇𝑘𝑔
  5. Problem: Harmonic Oscillator A mass 𝑚 attached to a spring with spring constant 𝑘 oscillates with an amplitude 𝐴. Find the maximum speed 𝑣max of the mass.

    • Given: 𝑚, 𝑘, 𝐴
    • Find: The maximum speed 𝑣max
    • Formula Involved: 𝐸total=12𝑘𝐴2=12𝑚𝑣max2

    Solution Steps:

    • Start with the energy conservation formula: 12𝑘𝐴2=12𝑚𝑣max2
    • Cancel out the 12 terms:
      𝑘𝐴2=𝑚𝑣max2
      𝑣max=𝐴𝑘𝑚

These problems are designed to make students use their algebraic skills to manipulate the given equations before arriving at the final answer.