Problem 1062 - JEE Physics

Question

If the length of a cylinder on heating increases by \(2 \%\) the area of its base will increase by. (A) \(0.5 \%\) (B) \(2 \%\) (C) \(1 \%\) (D) \(4 \%\)

Step-by-Step Solution

Answer

Since the radius of the cylinder's base remains the same after heating (r'= r), the area of the base also remains the same (A'= A = πr^2). Therefore, there is no increase in the area of the base, which means the correct answer is: (C) 0%

Step 1: Find the new radius after heating

Since the problem only mentions the increase in length of the cylinder, we can assume that the radius does not change when heated. Therefore, the radius before and after heating will be the same (r).

Step 2: Calculate the area of the base before and after heating

The area of the base before heating is given by the formula A = πr^2. Since the radius remains the same after heating, the area of the base after heating will also have the same formula A' = πr'^2. However, since r' = r, we have A' = πr^2.

Step 3: Compare the two base areas and find the percentage increase

Since the area of the base before and after heating is the same, A = A', there is no increase in the area of the base. The percentage increase in the area of the base would be: (\( \frac{A'-A}{A} \)) * 100% = (\( \frac{πr^2 - πr^2}{πr^2} \)) * 100% = 0% Therefore, the area of the base does not increase after heating. The correct option is: (C) 0%

Key Concepts

Cylinder Geometry

A cylinder is a three-dimensional geometric shape. It consists of two parallel circular bases connected by a curved surface. Each base has the same radius and is located at equal distance from the center, forming what is considered the height of the cylinder. The structure can be visualized as a circular column or pipe.

• The radius (r) of the base is the distance from the center of the circle to the edge.
• The height (h) is the distance between the two circular bases.
• The surface area includes both the areas of the two bases and the curved surface area connecting them.

Understanding the geometry of a cylinder is crucial when analyzing problems involving changes in dimension, such as thermal expansion. The base area of a cylinder can be calculated using the formula \( A = \pi r^2 \), where \( \pi \) (pi) approximates to 3.14159. The base area remains constant in cases where only the height is affected by factors such as temperature changes.

Percent Increase

The concept of percent increase describes how much a quantity grows in comparison to its original size, which is very important in understanding how changes may affect physical dimensions.
Percent increase is calculated by the formula:\[\text{Percent Increase} = \left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100\%\]In the context of the cylinder, if a measurement such as length increases by a certain percentage due to expansion, this does not mean every dimension changes proportionally.
For the given exercise, a 2% increase refers specifically to the height of the cylinder. Other dimensions, like the radius, may not necessarily change, affecting how the overall volume or surface area alters. Percent increase helps in quantifying these changes, which is essential for applications involving materials subjected to varying conditions.

Effect of Temperature on Materials

Temperature can significantly impact materials, especially those used in construction and mechanical applications. When materials are heated, they tend to expand; this phenomenon is called thermal expansion. It is particularly relevant for metal, plastic, wood, and other substances that may change dimensions when exposed to temperature variations.
In relation to cylinders, such as pipes and structural elements, thermal expansion primarily affects its length. Although the height of the cylinder may increase (as in the given exercise), the radius may stay unchanged, demonstrating anisotropic expansion characteristics. However, in materials that expand uniformly in all directions, both dimensions would change proportionately.
The rate and degree of expansion depend on several factors:

• Material type: Metals like aluminum and copper expand more than others like glass.
• Initial dimensions: Larger objects experience more noticeable expansion.
• Temperature change: Greater changes in temperature induce more expansion.

Understanding how temperature affects materials is crucial in designing and using cylinders effectively, ensuring their integrity and functionality in varying environmental conditions.