Classical Mechanics

Simple Harmonic Motion

Simple Harmonic Motion: Level 2-3 Challenges

         

In the figure two blocks are executing simple harmonic motion of same frequency in such a way that when one is at position of maximum displacement then other is at mean position. Now what is the path of external end point of rod.

When a ball with mass 1 kg1\text{ kg} is hung on a 50 cm50\text{ cm} long spring, the length of the spring is increased by 10 cm10\text{ cm} in steady state, as shown in the far left diagram. Now, suppose that we pull this ball further down and let it go. Then it will show simple harmonic oscillations. What will be the periods of these oscillations when we pull down the ball by 1010 cm and 2020 cm, respectively, as shown in the middle and right diagrams above?

Gravitational acceleration is 10 m/s210 \text{ m/s}^2.

What is the ratio of the period of a pendulum on Mars to the period of the same pendulum on Earth?

Details and assumptions

Gravity on Earth is 9.8 m/s29.8~m/s^{2}

Gravity on Mars is 3.71 m/s23.71~m/s^{2}

If the time period of simple harmonic motion of the rectangular block can be represented as T=2πambkT=2\pi \sqrt{\frac{am}{bk}} for coprime positive integers a,ba,b.

Find a2+b2\sqrt{a^2+b^2} up to two decimal places.

Details and Assumptions

  • Pulley is smooth and massless, and the strings are light and inextensible.
  • Gravity is present.
  • The crooked lines in the figure represents springs of spring constant kk and 2k2k as shown.

-Take the amplitude of oscillations to be very small.

A particle oscillating in simple harmonic motion has its motion timed at t=0st = 0 s when it is at the 50cm50 cm mark. It travels between the 50cm50 cm and 70cm70 cm marks with a period of 2.0s2.0 s. Where is the position of the particle at time t=0.75st = 0.75 s ?

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