The angular speed of a large pulley wheel that has a radius of 15, is 50 revolutions/min. Find the angular speed and the rim speed of the smaller wheel that has a radius of 5.
The rim speed is going to be the same.
I got the angular speed but I can't figure out why the rim speed is 4700 cm/min.
Please Help
rim speed=angular speed*radius
rim speed=50rev/min*2PIrad/1rev*15cm
= 4700cm/min
where did you get the 2Plrad?
To find the angular speed and rim speed of the smaller wheel, given the angular speed and radius of the larger wheel, we can use the concept of angular velocity.
We are given that the angular speed of the large pulley wheel is 50 revolutions per minute. To convert this into angular speed in radians per minute, we need to multiply it by 2π. This is because 1 revolution is equivalent to 2π radians. Thus, the angular speed of the large wheel can be calculated as follows:
Angular speed of the large wheel = 50 revolutions/min x 2π radians/revolution = 100π radians/min.
Next, we can use the formula for angular speed to relate the angular speeds of the two wheels. The formula is:
Angular speed = Rim speed / Radius of the wheel.
Let's represent the angular speed of the smaller wheel as ω1 and the rim speed as V1.
For the larger wheel:
Angular speed of the large wheel (ω2) = 100π radians/min.
Radius of the large wheel (r2) = 15 units.
For the smaller wheel:
Angular speed of the small wheel (ω1) = ?
Radius of the small wheel (r1) = 5 units.
Using the formula for angular speed, we can write:
ω1 = V1 / r1.
We know that the rim speed is the same for both wheels since they are in contact. Therefore, V1 = V2, where V2 represents the rim speed of the large wheel.
Let's calculate the rim speed of the large wheel first:
Rim speed of the large wheel (V2) = 50 revolutions/min x 2π x 15 units/revolution = 1500π units/min.
Since V1 = V2, we have:
Rim speed of the small wheel (V1) = 1500π units/min.
Now we can substitute the values of V1 and r1 into the equation for angular speed:
ω1 = (1500π units/min) / 5 units = 300π radians/min.
So, the angular speed of the small wheel is 300π radians/min.
Finally, to convert the angular rim speed into cm/min, we need to recall that the circumference of a circle is given by 2πr, where r is the radius. So, the rim speed (V) is calculated as:
V = ω × (2πr).
Plugging in the values, we have:
V = 300π radians/min × (2π × 5 cm) = 300 × 2π² × 5 cm/min = 300 × 20π² cm/min = 6000π² cm/min.
To get the numerical value of the rim speed, we can use an approximation for π:
V ≈ 6000 × (3.14)² cm/min = 6000 × 9.8596 cm/min ≈ 47100 cm/min.
Hence, the rim speed of the smaller wheel is approximately 47100 cm/min or 4700 cm/min, rounded to the nearest hundred.