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Chapter 17: Problem 19

A 2 -m-long shaft rotates about one end at \(20 \mathrm{rad} / \mathrm{s}\). It begins to accelerate with \(\alpha=10 \mathrm{rad} / \mathrm{s}^{2}\). After how long will the velocity of the free end reach \(100 \mathrm{~m} / \mathrm{s}\) ? a) \(3 \mathrm{~s}\) c) \(5 \mathrm{~s}\) b) \(4 \mathrm{~s}\) d) \(6 \mathrm{~s}\)

Short Answer

Expert verified
Answer: The time it takes for the free end to reach 100 m/s is 3 seconds.

Step by step solution

01

Find the initial linear velocity of the free end

We can find the initial linear velocity (v0) of the free end of the shaft using the formula: v0 = ω0 * r Where: ω0 = 20 rad/s (Initial angular velocity) r = 2 m (Length of the shaft) v0 = 20 rad/s * 2 m = 40 m/s The initial linear velocity of the free end of the shaft is 40 m/s.
02

Find the linear acceleration of the free end

Next, we find the linear acceleration (a) of the free end using the formula: a = α * r Where: α = 10 rad/s² (Angular acceleration) a = 10 rad/s² * 2 m = 20 m/s² The linear acceleration of the free end of the shaft is 20 m/s²
03

Find the time it takes for the free end to reach the desired velocity

Now, we can use the final linear velocity (vf), initial linear velocity (v0), and linear acceleration (a) to find the time (t) it takes for the free end to reach the desired velocity of 100 m/s. We can use the following equation: vf = v0 + a * t Solve for t: t = (vf - v0) / a Where: vf = 100 m/s (Final linear velocity) v0 = 40 m/s (Initial linear velocity) a = 20 m/s² (Linear acceleration) t = (100 m/s - 40 m/s) / 20 m/s² = 60 m/s / 20 m/s² = 3 s The time it takes for the free end to reach 100 m/s is 3 seconds. The correct answer is a) \(3 \mathrm{~s}\).

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