Angular Speed of Smaller Wheel

• MHB
In summary, the problem involves determining the angular speed of the smaller wheel, given the radius of both wheels and the angular speed of the larger wheel. The formula used is $r \omega_1 = R \omega_2$, where $\omega_1$ is the angular speed of the smaller wheel and $\omega_2$ is the angular speed of the larger wheel. After plugging in the given values, the angular speed of the smaller wheel is found to be $500/3$ rpm. Converting this to radians per minute, the final answer is $1000\pi/3$ radians per minute.

See picture.

If r = 6 cm, R = 10 cm, and the angular speed of the larger wheel is 100 rpm, determine the angular speed of the smaller wheel in radians per minute.

Again, 1 revolution = 2 pi radians.

I need to use w = θ/t.

So, θ = 100 rpm • 2 pi radians.

At this point, I doubted that my answer is right because it did not take long to find w or omega representing the angular speed.

What did I forget to do?

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f r = 6 cm, R = 10 cm, and the angular speed of the larger wheel is 100 rpm, determine the angular speed of the smaller wheel in radians per minute.

$v = r \omega_1$, where $\omega_1$ is the angular speed of the smaller wheel

$v = R \omega_2$, where $\omega_2$ is the angular speed of the larger wheel.

the belt connecting the two wheels has a fixed linear speed, $v$ ...

$\implies r \omega_1 = R \omega_2$

$\omega_1 = \dfrac{R \omega_2}{r} = \dfrac{10 \cdot 100}{6} = \dfrac{500}{3} \, rpm$

convert $\omega_1$ to radians per minute ...

$\dfrac{500 \, rev}{3 \, min} \cdot \dfrac{2\pi \, rad}{rev} = \dfrac{1000\pi}{3} \, rad/min$

Nicely-done. I will try your solution steps for this question to answer the textbook question where r = 6 cm and R = 10 cm.

1. What is the Angular Speed of a Smaller Wheel?

The angular speed of a smaller wheel refers to the rate at which the smaller wheel rotates around its center point. It is measured in radians per second (rad/s) or revolutions per minute (rpm).

2. How is Angular Speed of a Smaller Wheel calculated?

The angular speed of a smaller wheel can be calculated by dividing the angular displacement (change in angle) by the change in time. This can also be represented as the ratio of the circumference of the smaller wheel to the distance traveled by the center of the wheel in a given time period.

3. What factors affect the Angular Speed of a Smaller Wheel?

The angular speed of a smaller wheel can be affected by several factors including the size of the wheel, the torque applied, the friction in the wheel's bearings, and the rotational inertia of the wheel.

4. How does Angular Speed of a Smaller Wheel relate to Linear Speed?

The angular speed of a smaller wheel and the linear speed of a point on the wheel's edge are related through the formula v = rω, where v is the linear speed, r is the radius of the wheel, and ω is the angular speed. This shows that the linear speed increases as the angular speed or the radius of the wheel increases.

5. What are some real-life applications of Angular Speed of a Smaller Wheel?

The concept of angular speed of a smaller wheel is used in many real-life applications such as in the design of gears and gearboxes, calculating the speed of objects in circular motion such as wheels on a car or bike, and in understanding the movement of planets and satellites in space.

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