two gears are connected by a belt. the large gear has a radius of 20cm while the small gear has a radius of 10cm. if a point on the small gear travels at 16 rpm (revolutions per minute), find the angular velocity of the large gear. Tip: first find the velocity of the belt

same as ratio of circumferences

8 rpm = 8 * 2 pi radians/60 seconds

To find the angular velocity of the large gear, we need to first calculate the linear velocity of the belt.

The linear velocity of a point on the gear is given by the formula:
v = ω * r
where v represents linear velocity, ω represents angular velocity, and r represents the radius of the gear.

For the small gear, we are given its linear velocity, which is 16 rpm (revolutions per minute). Since we want to find the linear velocity of the belt, we need to calculate the linear velocity of a point on the small gear first.
v_small = ω_small * r_small

Given:
v_small = 16 rpm
r_small = 10 cm = 0.1 m

To convert from rpm (revolutions per minute) to rad/s (radians per second), we can use the following conversion factor:
1 rpm = 2π/60 rad/s

So, we can rewrite the equation as:
16 rpm = ω_small * 0.1 m

To solve for ω_small, we rearrange the equation:
ω_small = (16 rpm) / (0.1 m)

Now that we have the angular velocity of the small gear, we can determine the linear velocity of the belt, which is the same as the linear velocity of a point on the large gear.
v_large = ω_large * r_large

Given:
r_large = 20 cm = 0.2 m

To find ω_large, we use the fact that the linear velocities of both gears must be the same:
v_small = v_large
ω_small * r_small = ω_large * r_large

Now, we can substitute the known values:
(16 rpm) / (0.1 m) * 0.1 m = ω_large * 0.2 m

Simplifying the equation, we can cancel out the units:
16 rpm = 0.2 ω_large

To solve for ω_large, we rearrange the equation:
ω_large = (16 rpm) / (0.2)

Finally, we can calculate the angular velocity of the large gear:
ω_large = 16 rpm / 0.2
ω_large = 80 rpm

The angular velocity of the large gear is 80 rpm (revolutions per minute).