The speed of the skater's hands as she spins at 80.0 rpm with her arms 130 cm apart is approximately 5.45 m/s. To calculate this, the skater's rotational speed was first converted to radians per second, and then the linear velocity formula was applied using half the distance between her hands as the radius of rotation.
To find the speed of the hands of the skater, we will calculate the linear velocity at the point where her hands are located. Given that the skater rotates at 80.0 revolutions per minute (rpm), we first convert this rotational speed into revolutions per second (rps) because the standard unit of time for velocity is seconds. Next, we use the formula v = rω, where v is the linear velocity, r is the radius of rotation (half the distance between the hands), and ω is the angular velocity in radians per second.
Firstly, we convert the angular velocity:
80 rpm = ≈ 1.333 rps (since 1 minute = 60 seconds)
Since there are 2π radians in one revolution, we find the angular velocity in radians per second:
ω = 1.333 rps × 2π rad/rev = ≈ 8.38 rad/s
We then calculate the radius:
r = 130 cm / 2 = 65 cm = 0.65 m (since 1 m = 100 cm)
Now we can find the linear velocity of the skater's hands:
v = rω = 0.65 m × 8.38 rad/s = ≈ 5.45 m/s
Thus, the speed of the skater's hands is approximately 5.45 meters per second.