The formula ω = θ/t can be rewritten as θ = ωt. Substituting ωt for θ converts s = rθ to s = rωt. Use the formula s = rωt to find the value of the missing variable.

r = 6 cm, ω = π/3 radians per sec, t = 9 sec

검증된 단계별 안내

Identify the given variables: radius \(r = 6\) cm, angular velocity \(\omega = \frac{\pi}{3}\) radians per second, and time \(t = 9\) seconds.

Recall the formula relating arc length \(s\), radius \(r\), angular velocity \(\omega\), and time \(t\): \(s = r \omega t\).

Substitute the known values into the formula: \(s = 6 \times \frac{\pi}{3} \times 9\).

Simplify the expression step-by-step by first multiplying the constants and then including \(\pi\).

The result will give the arc length \(s\) in centimeters, which is the missing variable you need to find.

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주요 개념

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Angular Displacement and Angular Velocity

Angular displacement (θ) measures the angle through which an object rotates, typically in radians. Angular velocity (ω) is the rate of change of angular displacement over time, expressed in radians per second. The relationship ω = θ/t links these quantities, allowing calculation of one if the others are known.

Arc Length Formula

The arc length (s) of a circle segment is the distance traveled along the circumference and is given by s = rθ, where r is the radius and θ is the angular displacement in radians. This formula connects linear distance with angular motion, enabling conversion between rotational and linear measures.

Substitution to Relate Variables in Rotational Motion

By substituting θ = ωt into s = rθ, the formula becomes s = rωt, linking arc length directly to radius, angular velocity, and time. This substitution simplifies solving for any missing variable when the others are known, facilitating analysis of rotational motion problems.

The formula ω = θ/t can be rewritten as θ = ωt. Substituting ωt for θ converts s = rθ to s = rωt. Use the formula s = rωt to find the value of the missing variable.r = 6 cm, ω = π/3 radians per sec, t = 9 sec