# Angular speed of 2 pulleys on a belt

• MHB
• RobertoPink
In summary, the angular speed of 2 pulleys on a belt refers to the rate at which the pulleys rotate in a circular motion and is measured in radians per second. It can be calculated by dividing the linear speed of the belt by the radius of the pulley. Factors such as pulley size, belt tension, and external forces can affect the angular speed. This speed directly impacts the movement of the belt and has real-world applications in industries such as manufacturing and transportation. It is crucial for optimizing efficiency and ensuring the safe operation of systems that use pulleys and belts.
RobertoPink
The two pulleys connected by a belt have a radii of 15 cm and 8 cm. The larger pulley rotates 24 times in 36 seconds.

a. Find the angular velocity of the small pulley in radians per second.

b. Find the linear velocity of a point on the belt that connects the two pulleys in centimeters per second.

RobertoPink said:
The two pulleys connected by a belt have a radii of 15 cm and 8 cm. The larger pulley rotates 24 times in 36 seconds.

a. Find the angular velocity of the small pulley in radians per second.

b. Find the linear velocity of a point on the belt that connects the two pulleys in centimeters per second.

(a) $r \omega_1 = R\omega_2$

$r = 8 \, cm$, $R = 15 \, cm$, $\omega_2 = \dfrac{2}{3} \, \text{rev per sec}$

you'll need to convert $\omega_2$ to radians per second

(b) note $v = r \omega_1 = R\omega_2$

The larger pulley has radius 15 cm so circumference $2\pi(15)= 30\pi$ cm. The larger pulley rotates 24 times in 36 seconds so at a rate of 24/36= 2/3 rotations per second. In one second, since the larger pulley has rotated 2/3 of a rotation, the belt has moved a distance or $(2/3)(30\pi)= 20\pi$ cm.

The smaller pulley has radius 8 cm so circumference $2\pi(8)= 16\pi$. When the belt moves $20\pi$ cm, it will have moved $\frac{16\pi}{20\pi}= \frac{4}{5}$ of one rotation of the smaller pulley. The smaller pulley is turning at 4/5 rotation per second. There are $2\pi$ radians in one rotation so that is a rate of $(4/5)(2\pi)= (8/5)\pi$ radians per second.

The linear velocity of the belt is the $20\pi$ cm/sec we got earlier.

## What is the angular speed of 2 pulleys on a belt?

The angular speed of 2 pulleys on a belt refers to the rate at which the pulleys rotate in a circular motion. It is typically measured in radians per second.

## How is the angular speed of 2 pulleys on a belt calculated?

The angular speed of 2 pulleys on a belt can be calculated by dividing the linear speed of the belt by the radius of the pulley. This can also be expressed as the product of the number of rotations per unit time and 2π radians.

## What factors affect the angular speed of 2 pulleys on a belt?

The angular speed of 2 pulleys on a belt can be affected by the size and shape of the pulleys, the tension of the belt, and any external forces acting on the system. Friction between the belt and pulleys can also impact the angular speed.

## How does the angular speed of 2 pulleys on a belt impact the movement of the belt?

The angular speed of 2 pulleys on a belt directly affects the linear speed of the belt. As the pulleys rotate faster, the belt will move faster. This can also impact the tension and position of the belt on the pulleys.

## What are some real-world applications of understanding the angular speed of 2 pulleys on a belt?

Understanding the angular speed of 2 pulleys on a belt is important in many industries, such as manufacturing and transportation. It can be used to optimize the efficiency of machines and to ensure the safe and smooth movement of conveyor belts, elevators, and other systems that use pulleys and belts.

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