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.

s = 6π cm, r = 2 cm, ω = π/4 radian per sec

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Identify the given variables from the problem: arc length \(s = 6\pi\) cm, radius \(r = 2\) cm, and angular velocity \(\omega = \frac{\pi}{4}\) radians per second.

Recall the formula relating arc length, radius, and angular displacement: \(s = r\theta\). Since \(\theta = \omega t\), substitute to get \(s = r \omega t\).

Rearrange the formula \(s = r \omega t\) to solve for the missing variable \(t\): \(t = \frac{s}{r \omega}\).

Substitute the known values into the equation for \(t\): \(t = \frac{6\pi}{2 \times \frac{\pi}{4}}\).

Simplify the expression step-by-step to find the value of \(t\), which represents the time in seconds.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 along the curved path, calculated by s = rθ, where r is the radius and θ is the central angle in radians. This formula connects linear distance traveled along the circumference to angular displacement.

Combining Angular Velocity and Arc Length

By substituting θ = ωt into s = rθ, the formula becomes s = rωt, linking arc length directly to angular velocity and time. This allows solving for any missing variable (s, r, ω, or t) when the others are known, facilitating problems involving rotational motion.

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.s = 6π cm, r = 2 cm, ω = π/4 radian per sec