How Does Angular Velocity Affect Connected Gears?

[the_d]

Homework Statement

Gears A and B are connected by arm AB. The angular velocity at A is 60 rev. per minuter (clockwise), find the angular velocity at B if
a) wAB = 40 rpm (counterclockwise)
b) wAB = 40 rpm (clockwise)

Homework Equations

w=dTheta / dt
w = v/r

The Attempt at a Solution

solved for linear velocity for A (=480) do not know what to do next

 

[berkeman]

Is there a figure that goes with this question? What does it mean to connect two gear sprockets with an arm? Is it a toothed arm and A and B have diffeferent radii?

 

[kyle8921]

I would approach this problem by first understanding the concept of angular velocity and how it relates to gears. Angular velocity is the rate of change of angular displacement over time and is measured in radians per second. In the case of gears, the angular velocity of one gear is directly related to the angular velocity of the other gear it is connected to.

To solve this problem, we need to use the equation w = v/r, where w is the angular velocity, v is the linear velocity, and r is the distance from the center of rotation. In this case, we can assume that the distance from the center of rotation to the point where the gears are connected (arm AB) is the same for both gears.

a) If the angular velocity at A is 60 rpm (clockwise), we can calculate the linear velocity at A by using the formula v = w*r = (60 rpm)*(2*pi*r)/60 = 2*pi*r = 480. Since we know that the linear velocity at A is 480, we can use the same formula to calculate the angular velocity at B by rearranging the equation to w = v/r = (480)/r. If we assume that the distance from the center of rotation to B is the same as A, then the angular velocity at B would also be 60 rpm (counterclockwise).

b) Similarly, if the angular velocity at A is 60 rpm (clockwise), but the arm AB is rotating in the opposite direction, the linear velocity at A would still be 480, but the sign would be negative. This would result in a negative angular velocity at B, meaning it would also be rotating counterclockwise at 60 rpm.

In conclusion, the angular velocity of gears is directly related to their linear velocity and the distance from the center of rotation. By understanding this relationship, we can solve for the angular velocity at any point in the gear system.