Q1. Three identical bodies of mass m each are arranged as shown. What is the magnitude of the net gravitational force on Body 2?

Background

Topic: Newton's Law of Universal Gravitation

This question tests your understanding of gravitational forces between point masses and how to combine vector forces from multiple sources.

Key Terms and Formulas:

• Newton's Law of Universal Gravitation:

• Where is the gravitational constant, and are the masses, and is the distance between them.

Step-by-Step Guidance

1. Identify the forces acting on Body 2 due to the other two bodies. Each force will have a magnitude , where is the distance between the bodies.

2. Determine the direction of each force. Since the bodies are arranged perpendicularly (as implied by the solution), one force acts along the -axis and the other along the -axis.

3. Express the net force as the vector sum of these two perpendicular forces.

4. Use the Pythagorean theorem to find the magnitude of the net force: .

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Q2. If you throw a rock vertically upward from the ground at half the escape speed, what is the maximum altitude (distance from the ground) it reaches? Express your answer in terms of Earth's radius .

Background

Topic: Conservation of Energy in Gravitational Fields

This question tests your ability to apply energy conservation to gravitational systems, specifically using the concept of escape velocity and gravitational potential energy.

Key Terms and Formulas:

• Escape velocity:

• Gravitational potential energy:

• Kinetic energy:

• Conservation of energy:

Step-by-Step Guidance

1. Write the total mechanical energy at launch: , where .

2. At maximum altitude, the rock's velocity is zero, so , where is the altitude above Earth's surface.

3. Set and substitute and into the equation.

4. Rearrange the equation to solve for in terms of .

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Q3. A velocity versus time graph of a body undergoing simple harmonic motion is shown. At which moments is the body at the center of oscillation and moving in the positive direction?

Background

Topic: Simple Harmonic Motion (SHM)

This question tests your understanding of the relationship between velocity, position, and direction of motion in SHM, especially interpreting velocity-time graphs.

Key Terms and Concepts:

• Center of oscillation: The equilibrium position ().

• Maximum speed: Occurs as the object passes through equilibrium.

• Positive direction: When velocity is positive (above the time axis).

Step-by-Step Guidance

1. Recall that in SHM, the object passes through the center (equilibrium) when its velocity is at a maximum (positive or negative).

2. Identify the points on the velocity-time graph where velocity is at a positive maximum.

3. Match these points to the labeled moments (e.g., 1, 2, 3, 4) on the graph.

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Q4. (i) A cilium in the human respiratory tract is modeled as a simple harmonic oscillator. At , the tip is at equilibrium; four seconds later, it is at its maximum displacement . What is the equation for its position as a function of time?

Background

Topic: Simple Harmonic Motion – Mathematical Description

This question tests your ability to write the equation of motion for a simple harmonic oscillator given initial conditions and timing information.

Key Terms and Formulas:

• General SHM equation: or

• Period (): Time for one complete oscillation.

• Angular frequency:

Step-by-Step Guidance

1. Since the cilium is at equilibrium at , consider which function ( or ) fits this initial condition.

2. Recognize that reaching maximum displacement after 4 seconds means s corresponds to (maximum amplitude).

3. Relate this timing to the period: from equilibrium to maximum is one-quarter of a period, so s.

4. Write the equation for using the amplitude , the correct trigonometric function, and the calculated period.

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Q4. (ii) Where is the magnitude of the force on the cilium's tip greatest?

Background

Topic: Simple Harmonic Motion – Force and Acceleration

This question tests your understanding of where the restoring force is maximized in SHM.

Key Terms and Formulas:

• Restoring force in SHM:

• The magnitude is greatest when is greatest (i.e., at maximum displacement).

Step-by-Step Guidance

1. Recall that the force is proportional to displacement from equilibrium: .

2. The magnitude of force is maximized when is maximized, i.e., at and .

3. Consider the answer choices and select those corresponding to maximum displacement.

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