Wheel Making 360 Revolutions per Minute — Radians Turned in 1 Second

NCERT Class 11 Mathematics — Trigonometric Functions, Exercise 3.1, Question 3.

Problem

A wheel makes $360$ revolutions in $1$ minute. Find the number of radians it turns through in $1$ second.

Key insight. The problem gives a rate per minute but asks for the result per second — converting the time unit first, before touching the revolutions-to-radians conversion, keeps the arithmetic clean.

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Approach

Convert the given rate from revolutions per minute to revolutions per second, then convert each revolution into its radian equivalent using the standard fact that one full revolution is $2\pi$ radians.

Solution

Step 1 — Convert revolutions per minute to revolutions per second

Since $1$ minute $= 60$ seconds, and the wheel makes $360$ revolutions in that time:

$$\text{Revolutions per second} = \frac{360}{60} = 6$$

Step 2 — Convert revolutions to radians

One full revolution corresponds to $2\pi$ radians, so:

$$\text{Radians per second} = 6 \times 2\pi = 12\pi$$

Answer

Common mistakes

• Converting revolutions to radians before converting minutes to seconds. Doing the unit conversion in the wrong order doesn’t change the final answer here, but it’s easy to lose track of which number represents which unit if the steps are combined carelessly.
• Forgetting that one revolution is $2\pi$ radians, not $\pi$ radians. A full circle is $360°=2\pi$ radians; $\pi$ radians is only a half revolution ($180°$).

Practise next

• A wheel makes $900$ revolutions in $2$ minutes. Find how many radians it turns through in $1$ second, using the same two-step conversion.