The angular speed of a larger pulley wheel (radius=15cm) is 50 rev./min. Find the angular speed of the smaller wheel (radius=6cm).
Assuming the two wheels are linked by a belt, then the tangential velocities are equal, i.e. the products of angular speed and radius are equal:
ω1(r1)=ω2(r2), or
ω2=ω1(r1/r2)
125rev/min
To find the angular speed of the smaller wheel, we can use the formula:
ω₁ = (r₂ / r₁) * ω₂
Where:
ω₁ = angular speed of the smaller wheel
ω₂ = angular speed of the larger wheel
r₁ = radius of the smaller wheel
r₂ = radius of the larger wheel
Given:
ω₂ = 50 rev/min
r₁ = 6 cm
r₂ = 15 cm
Plugging in the known values into the formula, we have:
ω₁ = (15 cm / 6 cm) * 50 rev/min
Simplifying:
ω₁ = (2.5) * 50 rev/min
ω₁ = 125 rev/min
Therefore, the angular speed of the smaller wheel is 125 rev/min.
To find the angular speed of the smaller wheel, we can use the concept of angular velocity, which is defined as the rate at which an object rotates about an axis. The angular velocity is usually measured in radians per second (rad/s).
Here's how you can solve the problem step by step:
1. Convert the units: In the given question, the angular speed of the larger wheel is given in revolutions per minute (rev/min). To find the angular speed of the smaller wheel, we need to convert this to radians per second (rad/s).
Since 1 revolution is equal to 2π radians, we can convert revolutions per minute to radians per second using the following conversion factor:
1 rev/min = 2π rad/60 s
2. Calculate the angular velocity of the larger wheel: The angular speed of the larger wheel is given as 50 rev/min. We can calculate its angular velocity by multiplying the angular speed by the conversion factor:
Angular velocity (ω) = 50 rev/min × (2π rad/60 s)
3. Calculate the linear velocities of the larger and smaller wheels: The linear velocity of a point on a rotating object is given by the product of its angular velocity and the radius of the object.
For the larger wheel:
Linear velocity = Angular velocity × Radius
Linear velocity of the larger wheel = ω × 15 cm
4. Calculate the angular velocity of the smaller wheel: Since the linear velocities of both wheels are equal, we can set up the following equation:
Linear velocity of the larger wheel = Linear velocity of the smaller wheel
Using the equation Linear velocity = Angular velocity × Radius, we can write:
ω × 15 cm = Linear velocity of the smaller wheel = ω' × 6 cm
Simplifying the equation, we find that:
ω' = (ω × 15 cm) / 6 cm
Now, substitute the value of ω calculated in step 2 into the above equation to find the angular velocity of the smaller wheel.
By following these steps, you can find the angular speed of the smaller wheel (radius = 6 cm) using the given angular speed of the larger wheel (radius = 15 cm) in revolutions per minute.