Question

Water raises in a vertical capillary tube upto a height of \(2.0\) \(\mathrm{cm}\). If tube is inclined at an angle of \(60^{\circ}\) with the vertical then the what length the water will rise in the tube. (A) \(2.0 \mathrm{~cm}\) (B) \(4.0 \mathrm{~cm}\) (C) \((4 / \sqrt{3}) \mathrm{cm}\) (D) \(2 \sqrt{2} \mathrm{~cm}\)

Short answer

The length of the water rise in the inclined capillary tube is \(4.0 \mathrm{~cm}\) (Option (B)).

Step-by-step solution

1. Identifying known and unknown variables

The given information is that in the vertical position, the water has risen to a height of 2.0 cm in the capillary tube. The angle of inclination (\(\theta\)) of the tube is 60 degrees. We need to find the length of the water rise in the tube when it's inclined.

2. Finding the relationship between the variables

In order to find the length of the water rise in the tube when it's inclined, we will use trigonometry. Consider a right-angled triangle formed by the inclined tube, the vertical height of the water rise, and the horizontal distance from the base of the tube. In the vertical position, the capillary action is counteracting gravity, so this force balance remains the same when the tube is inclined. Therefore, the vertical height of the water rise (\(h\)) remains constant at 2.0 cm when the tube is inclined.

3. Using trigonometry to relate the variables

Using the trigonometric relationship between the angle and the sides of the right-angled triangle, we have: \(\cos \theta = \frac{h}{l}\), Where \(\theta\) is the angle of inclination (60 degrees), \(h\) is the constant vertical height of the water rise (2.0 cm), and \(l\) is the length of the water rise in the inclined tube (our unknown).

4. Solving for the unknown variable

We need to solve the equation for the unknown variable \(l\). Using the given angle of 60 degrees, the equation becomes: \(\cos 60^{\circ} = \frac{2.0 \mathrm{~cm}}{l}\), And we know, \(\cos 60^{\circ} = \frac{1}{2}\), So, \(\frac{1}{2} = \frac{2.0 \mathrm{~cm}}{l}\), Now, we can solve for \(l\): \(l=\frac{2.0 \mathrm{~cm}}{\frac{1}{2}} = 4.0 \mathrm{~cm}\).

5. Final answer

The length of the water rise in the inclined capillary tube is \(4.0 \mathrm{~cm}\) (Option (B)).

Key concepts

Trigonometry in Physics

In physics, trigonometry is a powerful tool, especially when analyzing forces, motions, and angles in systems. In the capillary action scenario, where water rises due to the attraction between the liquid and solid surfaces, trigonometry helps us understand how this behavior changes when the system's orientation is adjusted.

By breaking down the system into measurable elements, like angles and vertical rises, we can apply trigonometric ratios to gain insights. These ratios include sine, cosine, and tangent, which relate the angles in a right triangle to the length of its sides. Here, the cosine function (\( \cos \theta = \frac{h}{l} \)) illustrates how much the rise length changes with the tube's angle to gravity.

Force Balance

Force balance is an essential concept when dealing with capillary action. It refers to the equilibrium of forces acting on the fluid within the tube. When a tube is perfectly vertical, the upward capillary force directly opposes gravity, resulting in a measurable rise.

Inclining the tube does not alter the magnitude of these forces but changes the direction relative to the tube. The vertical component of the force remains constant, ensuring the height the water reaches doesn't change. This consistency allows us to use trigonometric relationships to determine the new rise length along the tube when inclined.

Angle of Inclination

The angle at which an object is inclined can significantly impact calculations in physics. It's defined in this context as the angle between the vertical line and the axis of the inclined tube.

Applying this to capillary action, the angle (\( \theta \)) influences how we calculate the length the water will reach in the inclined tube. By understanding this angle, we adjust our mathematical model to consider how the vertical rise projects onto the inclined tube's length. Utilizing \( \cos 60^{\circ} \) provided known values, it guides us to find the extended reach of the fluid in such settings.

Right-Angled Triangle

In the inclined capillary tube scenario, imagining a right-angled triangle is useful. One side represents the vertical rise of the water, while another side represents the inclined tube's length. The angle at which the tube inclines forms the right triangle's focus, with the hypotenuse (the longest side) being the unknown variable we aim to find.

Utilizing the properties of right-angled triangles, we use the cosine function to tie together the known vertical rise with the inclined length. Through \( \cos \theta = \frac{h}{l} \), we solve for \( l \), the hypotenuse, showing how understanding simple geometric shapes can explain physical phenomena like capillary action in inclined systems.