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 6.00 Hz counter clockwise, what is B’s angular velocity? (b) If A’s radius to the tip of the teeth is 10.0 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?

Think about it. The speed of the gears' boundaries must match. Since A has a larger radius, it turns more slowly in rpm.

So, if A is 6Hz ccw, B is 12Hz cw.

The linear speed must match,or the teeth will strip!

A's speed is 2π(10)*6 = 120πcm/s
B's speed is the same: 2π(5)*12

To solve this problem, we need to use the concept of rotational motion and the relationship between linear and angular velocities in gears.

(a) To find gear B's angular velocity when gear A's angular velocity is given, we need to apply the principle of conservation of angular momentum. Angular momentum is conserved when two gears are in contact and their radii are different.

The relationship between the angular velocities of two gears in contact is:

Angular velocity of A / Angular velocity of B = Radius of B / Radius of A

Let's denote the angular velocity of gear B as ωB. Given that the radius of gear A is twice that of gear B, we can write:

6.00 Hz (angular velocity of A) / ωB = 1/2

Now, we can solve for ωB:

ωB = (6.00 Hz) / (1/2)
= (6.00 Hz) * (2/1)
= 12.00 Hz

Therefore, the angular velocity of gear B when gear A's angular velocity is 6.00 Hz counterclockwise is 12.00 Hz.

(b) To find the linear speed of a point on the tip of the gear tooth, we can use the relationship between linear and angular velocities.

The linear speed of a point on the tip of a gear tooth is given by:

Linear speed = Radius * Angular velocity

For gear A, given that the radius to the tip of the teeth is 10.0 cm, we can calculate its linear speed:

Linear speed of A = (10.0 cm) * (6.00 Hz)
= 60.0 cm/s

For gear B, since the radius of B is half that of A, its linear speed can be calculated as:

Linear speed of B = (5.0 cm) * (12.00 Hz)
= 60.0 cm/s

Therefore, the linear speed of a point on the tip of the tooth for both gears A and B is 60.0 cm/s.