Q1. The position of a 100 g mass oscillating on a massless spring is given by . Identify the following:

Background

Topic: Simple Harmonic Motion (SHM)

This question explores the properties of a mass-spring system undergoing simple harmonic motion, including amplitude, period, frequency, angular frequency, energy, and the relationship between kinetic and potential energy.

Key Terms and Formulas

• Amplitude (): Maximum displacement from equilibrium.

• Period (): Time for one complete oscillation.

• Frequency (): Number of oscillations per second, .

• Angular frequency (): .

• Total energy (): .

• Spring constant (): .

• Kinetic energy () at position : .

• Potential energy () at position : .

• Speed at position : .

Step-by-Step Guidance

1. Identify the amplitude (): The amplitude is the coefficient in front of the cosine function in the position equation. Check the equation and note the value of .

2. Find the period (): The period is related to the argument of the cosine. The general form is , but here it's . Match this to the given equation to identify .

3. Calculate the frequency (): Use to find the frequency in Hz.

4. Determine the angular frequency (): Use or .

5. Find when for the first time: Set in the position equation and solve for (the first positive value).

6. Calculate the total energy (): Use . You'll need to find first (see below).

7. Find the spring constant (): Use , where is the mass in kg and is the angular frequency.

8. Calculate the speed at : Use , making sure all units are consistent (convert cm to m if needed).

9. Does the speed at depend on direction? Consider whether the formula for speed depends on the sign of velocity or just its magnitude.

10. Find the kinetic energy at : .

11. Find the potential energy at : .

12. Sum the kinetic and potential energies at : Add the values from the previous two steps.

13. Compare the total energy and the sum of kinetic and potential energies: Reflect on the principle of conservation of energy in SHM.

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Q2. A 3.8-kg mass on a massless spring is set into oscillatory motion from with an initial velocity in the positive x-direction. The motion is described by a sine curve: . Answer the following:

Background

Topic: Simple Harmonic Motion (SHM) with Sine Initial Condition

This question examines SHM when the initial position is zero and the initial velocity is nonzero, leading to a sine function description. It covers amplitude, period, frequency, force, speed, and the relationship between force and velocity.

Key Terms and Formulas

• Amplitude (): Maximum displacement from equilibrium.

• Period (): Time for one complete oscillation.

• Frequency (): .

• Angular frequency (): .

• Force on mass: (maximum when is maximum).

• Speed: (maximum when ).

• At maximum force, speed is zero (turning points); at minimum force (), speed is maximum.

Step-by-Step Guidance

1. Find the amplitude (): Read the maximum value of from the graph or equation.

2. Determine the period (): Measure the time for one complete cycle from the graph or use the equation.

3. Calculate the frequency (): Use .

4. Find when the force is maximum between 2 and 4 s: Force is maximum when is maximum (peaks of the sine curve). Identify these times from the graph.

5. Find the speed when force is maximum: At maximum , the mass changes direction, so speed is zero.

6. Find when force is minimum between 1.5 and 2.5 s: Force is minimum (zero) when (crosses equilibrium).

7. At minimum force, is speed minimum or maximum? At , speed is maximum in SHM.

8. Write the equation with values: Use amplitude and period to find and .

9. Find when m for the first time: Set m and solve for using the sine equation.

10. Find when m for the second time: Use the periodicity of the sine function to find the next value. Compare with the graph for confirmation.

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Q3. A 0.95-kg mass attached to a vertical spring of force constant 130 N/m oscillates with a maximum speed of 0.45 m/s. Find:

• a) The period of oscillation

• b) The amplitude of the motion

• c) The maximum magnitude of the acceleration

Background

Topic: Energy and Kinematics in SHM

This question tests your ability to relate mass, spring constant, amplitude, speed, and acceleration in a mass-spring system.

Key Terms and Formulas

• Period:

• Maximum speed:

• Angular frequency:

• Amplitude:

• Maximum acceleration:

Step-by-Step Guidance

1. Calculate the angular frequency ():

2. Find the period ():

3. Determine the amplitude (): Use and solve for .

4. Calculate the maximum acceleration ():

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Q4. A student lowers a rock on a rope into a deep vertical hole and sets it into simple harmonic motion. The rock takes 233 seconds to complete 10 oscillations. How deep is the hole?

Background

Topic: Simple Pendulum Period and Length

This question uses the period of a simple pendulum to determine the length (depth) of the hole, assuming the rope acts as a simple pendulum.

Key Terms and Formulas

• Period of a simple pendulum:

• Total time for oscillations:

• Acceleration due to gravity:

Step-by-Step Guidance

1. Find the period () for one oscillation: , where is total time and is number of oscillations.

2. Set up the pendulum period formula:

3. Rearrange to solve for :

4. Plug in the values for and to find (the depth of the hole).

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