Problem 19 - University Physics with Modern Physics

Question

If you turn on the faucet in the bathroom sink, you will observe that the stream seems to narrow from the point at which it leaves the spigot to the point at which it hits the bottom of the sink. Why does this occur?

Step-by-Step Solution

Answer

Answer: The stream of water narrows because of the conservation of mass and the effect of gravity. As the water accelerates due to gravity, its velocity increases, causing the cross-sectional area of the stream to decrease to maintain constant mass flow.

Step 1: 1. Understand the basic setup

We need to analyze the flow of water from the faucet to the sink to understand why the stream narrows. The water flows through the spigot, becoming narrower as it falls to the sink bottom.

Step 2: 2. Introduce the conservation of mass

The conservation of mass states that the mass of an object remains constant over time, no matter how it may change in shape or volume. For our water stream, the mass of water entering the spigot must equal the mass of water exiting and hitting the sink bottom.

Step 3: 3. Apply the continuity equation to the water stream

The continuity equation is a fundamental principle in fluid dynamics that relates the velocity and cross-sectional area of the flow at different points along the stream. The equation states that the product of the cross-sectional area (A) and the velocity (v) at any point along the stream is constant: \\ A_1 v_1 = A_2 v_2 , \\ where A_1 and v_1 represent the area and velocity at the beginning of the stream (near the faucet), and A_2 and v_2 represent the area and velocity near the sink bottom.

Step 4: 4. Explain the effect of gravity on the water stream

Due to gravity, water accelerates as it falls from the faucet to the sink bottom. An increase in velocity (from v_1 to v_2) would cause the velocity near the sink bottom to be higher than near the faucet.

Step 5: 5. Derive the relationship between cross-sectional areas

Since the product of area and velocity is constant (from the continuity equation), if the velocity increases, the cross-sectional area must decrease to maintain the equation balance. Therefore, the cross-sectional area near the sink (A_2) would be smaller than the area near the faucet (A_1): A_2 < A_1.

Step 6: 6. Conclude the explanation

The stream of water from the faucet to the sink bottom narrows as it falls because, as gravity accelerates the water, the velocity increases. To maintain constant mass flow due to the conservation of mass, the cross-sectional area of the stream must decrease. Consequently, the water stream appears to narrow from the faucet to the point where it hits the sink.