Understanding Plane Angle from Arc and Radius

The question asks us to identify the physical quantity represented by the expression arc/radius. This ratio is a fundamental concept in geometry and physics used to define a specific type of angle.

Defining Plane Angle

A plane angle is a measure of the rotation between two intersecting lines or surfaces. When we consider a circle, a plane angle subtended at the center can be defined using the arc length and the radius.

The formula that relates the arc length (\(s\)), the radius (\(r\)), and the plane angle (\(\theta\)) is:

\(\theta = \frac{s}{r}\)

Here:

• \(s\) is the length of the arc along the circumference.
• \(r\) is the radius of the circle.
• \(\theta\) is the plane angle subtended by the arc at the center.

This formula defines the plane angle specifically in units called radians. A radian is the angle subtended at the center of a circle by an arc that has a length equal to the radius.

Analyzing the Options

Let's look at the given options to see which one matches the expression arc/radius:

1. Linear momentum: Linear momentum (\(p\)) is defined as the product of mass (\(m\)) and velocity (\(v\)): \(p = mv\). Its units are typically kilogram meters per second (kg·m/s). This is clearly not represented by the ratio of arc length to radius.
2. Velocity: Velocity (\(v\)) is the rate of change of displacement. Its units are typically meters per second (m/s). This is also not represented by the ratio of arc length to radius.
3. Plane angle: As discussed above, the plane angle (\(\theta\)) subtended by an arc at the center of a circle is defined as the ratio of the arc length (\(s\)) to the radius (\(r\)), i.e., \(\theta = s/r\). When \(s\) and \(r\) are measured in the same unit of length (like meters), the ratio \(s/r\) is dimensionless. Radians are the SI unit for plane angle, often considered a dimensionless derived unit or a base unit depending on the convention.
4. Surface tension: Surface tension (\(\gamma\)) is a property of liquid surfaces that describes the force per unit length or energy per unit area. Its units are typically Newtons per meter (N/m) or Joules per square meter (J/m²). This is unrelated to the ratio of arc length to radius.

Based on the definition and analysis of the options, the expression arc/radius represents the physical quantity known as the plane angle.

Conclusion

The expression arc/radius is the mathematical definition of the plane angle in radians. Therefore, the physical quantity is plane angle.

Revision Table: Key Physical Quantities

Additional Information on Angles and Units

While the ratio arc/radius defines the plane angle in radians, angles can also be measured in other units like degrees. The relationship between radians and degrees is important:

• A full circle is \(2\pi\) radians or 360 degrees.
• \(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\)
• \(1 \text{ degree} = \frac{\pi}{180} \text{ radians}\)

The concept of angle can be extended to three dimensions, where it is called a solid angle. A solid angle is measured in steradians (sr) and is defined as the ratio of the surface area subtended on a sphere to the square of the sphere's radius (\(\Omega = A/r^2\)). The plane angle (arc/radius) is the 2D analog of the solid angle (area/radius²).

Understanding the units and definitions of physical quantities is crucial in physics problems. Always pay attention to the dimensions involved in expressions to help identify the physical quantity.