Q1. What is the ratio of the masses of the two blocks mA : mB, given that standing waves of identical frequencies are formed on identical ropes, each with tension provided by a hanging block?

Background

Topic: Waves on a String

This question tests your understanding of how the speed of a wave on a string depends on the tension and mass per unit length, and how tension is related to the mass of the hanging block.

Key formula:

Where:

• = speed of the wave

• = tension in the rope (provided by the weight of the block, )

• = mass per unit length of the rope

Step-by-Step Guidance

1. Recall that the speed of the wave depends on the tension and mass per unit length: .

2. Since the ropes are identical and the frequencies are identical, the wave speeds must be equal for both ropes.

3. Set the wave speeds equal: .

4. Since is the same for both ropes, this simplifies to .

5. Recall that tension is provided by the hanging block: , so and .

Try solving on your own before revealing the answer!

Q2. A uniform block floats with 1/3 of its volume above the surface of water. Find the mass density of the block in terms of the mass density of water .

Background

Topic: Buoyancy and Archimedes' Principle

This question tests your understanding of how the fraction of a floating object's volume above water relates to its density compared to the fluid.

Key formula:

Where:

• = density of the block

• = density of water

Step-by-Step Guidance

1. Let be the total volume of the block. is the submerged volume.

2. Since 1/3 of the block is above water, 2/3 is submerged: .

3. Apply Archimedes' Principle: the weight of the block equals the weight of displaced water.

4. Set up the equation: .

5. Substitute and solve for in terms of .

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Q3. A Doppler ultrasound device emits waves at MHz. Red blood cells move toward the transducer at m/s. The speed of sound in tissue is m/s. What is the frequency of the reflected wave detected by the transducer?

Background

Topic: Doppler Effect for Sound Waves

This question tests your ability to apply the Doppler effect for waves reflected from moving objects, specifically in a medical context.

Key formula:

(for moving observer)

For reflection, apply the Doppler effect twice: once for the moving blood cells as observer, then as source.

Where:

• = source frequency

• = speed of sound in tissue

• = speed of blood cells toward the source

Step-by-Step Guidance

1. First, calculate the frequency as received by the moving blood cells (observer): .

2. Next, the blood cells act as a moving source reflecting the wave back: .

3. Combine the two steps: .

4. Plug in the values: MHz, m/s, m/s.

5. Set up the calculation, but do not compute the final value yet.

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Q4. Water passes steadily through a pipe where the cross-sectional area at A is twice as big as at B. At which point is the volume flow rate greater, and by how much?

Background

Topic: Fluid Dynamics & Continuity Equation

This question tests your understanding of the continuity equation for incompressible fluids, which relates flow rate to cross-sectional area and velocity.

Key formula:

Where:

• = volume flow rate

• = cross-sectional area

• = fluid velocity

Step-by-Step Guidance

1. Recall the continuity equation: for incompressible fluids, .

2. Set up the equation: .

3. Given , relate the velocities: .

4. Calculate the flow rates at each point: , .

5. Compare and using the relationships above, but do not compute the final comparison yet.

Try solving on your own before revealing the answer!