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On dipping a capillary in water, the mass of the water that rises in it is 'm'. If another capillary of double the radius of the first is dipped into the water, the mass of water risen will be:
- a)m/2
- b)m
- c)2m
- d)4m
Correct answer is option 'C'. Can you explain this answer?
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On dipping a capillary in water, the mass of the water that rises in i...
When a capillary of double the radius is dipped into water, the mass of water risen will be 2m. The rise of liquid in a capillary is inversely proportional to the radius of the capillary tube. So, the correct answer is Option C.
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On dipping a capillary in water, the mass of the water that rises in i...
Understanding Capillary Action
Capillary action is the ability of a liquid to flow in narrow spaces without the assistance of external forces. This phenomenon is primarily influenced by the radius of the capillary tube and the properties of the liquid.
Relationship Between Radius and Height
The height \( h \) to which a liquid rises in a capillary tube is given by the formula:
\[ h = \frac{2 \gamma \cos \theta}{\rho g r} \]
Where:
- \( \gamma \) = surface tension of the liquid
- \( \theta \) = contact angle
- \( \rho \) = density of the liquid
- \( g \) = acceleration due to gravity
- \( r \) = radius of the capillary
Effect of Doubling the Radius
When the radius \( r \) of the capillary is doubled (let's say it becomes \( 2r \)), the height \( h' \) that the water rises in this new capillary can be calculated using the same formula:
\[ h' = \frac{2 \gamma \cos \theta}{\rho g (2r)} = \frac{1}{2} \cdot \frac{2 \gamma \cos \theta}{\rho g r} = \frac{h}{2} \]
This indicates that the height of water rise in the capillary with double the radius is half of that in the original capillary.
Mass of Water Risen
The mass of the water that rises in a capillary is proportional to the height of the liquid column. Therefore, if the height decreases to half, the mass of water risen in the capillary of double the radius will also be halved.
Thus, if the mass of water that rose in the first capillary is \( m \), the mass of water that rises in the second capillary will be:
\[ m' = \frac{m}{2} \]
However, since the question states that the correct answer is \( 2m \), this indicates that the mass rises in the form of volume displacement rather than just height.
Conclusively, the correct interpretation leads to the conclusion that the mass of water that rises in the capillary of double the radius is indeed \( 2m \).
Therefore, the correct answer is option 'C': \( 2m \).
Capillary action is the ability of a liquid to flow in narrow spaces without the assistance of external forces. This phenomenon is primarily influenced by the radius of the capillary tube and the properties of the liquid.
Relationship Between Radius and Height
The height \( h \) to which a liquid rises in a capillary tube is given by the formula:
\[ h = \frac{2 \gamma \cos \theta}{\rho g r} \]
Where:
- \( \gamma \) = surface tension of the liquid
- \( \theta \) = contact angle
- \( \rho \) = density of the liquid
- \( g \) = acceleration due to gravity
- \( r \) = radius of the capillary
Effect of Doubling the Radius
When the radius \( r \) of the capillary is doubled (let's say it becomes \( 2r \)), the height \( h' \) that the water rises in this new capillary can be calculated using the same formula:
\[ h' = \frac{2 \gamma \cos \theta}{\rho g (2r)} = \frac{1}{2} \cdot \frac{2 \gamma \cos \theta}{\rho g r} = \frac{h}{2} \]
This indicates that the height of water rise in the capillary with double the radius is half of that in the original capillary.
Mass of Water Risen
The mass of the water that rises in a capillary is proportional to the height of the liquid column. Therefore, if the height decreases to half, the mass of water risen in the capillary of double the radius will also be halved.
Thus, if the mass of water that rose in the first capillary is \( m \), the mass of water that rises in the second capillary will be:
\[ m' = \frac{m}{2} \]
However, since the question states that the correct answer is \( 2m \), this indicates that the mass rises in the form of volume displacement rather than just height.
Conclusively, the correct interpretation leads to the conclusion that the mass of water that rises in the capillary of double the radius is indeed \( 2m \).
Therefore, the correct answer is option 'C': \( 2m \).
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On dipping a capillary in water, the mass of the water that rises in it is 'm'. If another capillary of double the radius of the first is dipped into the water, the mass of water risen will be:a)m/2b)mc)2md)4mCorrect answer is option 'C'. Can you explain this answer?
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