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Chapter 5: Problem 68

Two gears \(A\) and \(B\) are in contact. The radius of gear \(A\) is twice that of gear \(B\). (a) When \(A\) 's angular velocity is \(6.00 \mathrm{Hz}\) counterclockwise, what is \(B^{\prime}\) s angular velocity? (b) If \(A\) 's radius to the tip of the teeth is \(10.0 \mathrm{cm},\) what is the linear speed of a point on the tip of a gear tooth? What is the linear speed of a point on the tip of \(B\) 's gear tooth?

Short Answer

Expert verified
Answer: The linear speed of a point on the tip of Gear B's gear tooth is \(1.2\pi \mathrm{m/s}\).

Step by step solution

01

Understand the relationship between gears A and B and find a ratio

Since the radius of gear A is twice that of gear B, their circumferences have the same ratio because the circumference is directly proportional to the radius. The ratio is 2:1 and since the gears are in contact, they must have the same linear velocity.
02

Calculate angular velocity of gear B

Angular velocity of gear A (\(\omega_A\)) is given as \(6.00 \mathrm{Hz}\). To find the angular velocity of gear B (\(\omega_B\)), we divide gear A's angular velocity by the ratio (2:1) found in the previous step. $$ \omega_B = \frac{6.00 \mathrm{Hz}}{\frac{2}{1}} = 3.00 \mathrm{Hz} $$
03

Calculate the linear speed of the tip of Gear A

Now that we found the angular velocity of Gear A, we can calculate the linear speed (\(v_A\)) of a point on the tip of the gear tooth. We use the following equation: $$ v_A = r_A \cdot \omega_A $$ where, \(r_A\) is the radius of Gear A, and \(\omega_A = 6.00 \mathrm{Hz}\) Given Gear A's radius is \(10.0 \mathrm{cm}\) or \(0.1 \mathrm{m}\), we can calculate the linear speed: $$ v_A = 0.1 \mathrm{m} \cdot 6.00 \text{ 2}\pi \mathrm{rad/s} = 1.2\pi \mathrm{m/s} $$
04

Calculate the linear speed of the tip of Gear B

Since gears A and B are in contact and have the same linear velocity (as found in step 1), the linear speed of a point on the tip of Gear B's gear tooth is equal to the linear speed of Gear A which is \(1.2\pi \mathrm{m/s}\).

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