Intro to Connected Wheels: Videos & Practice Problems

In problems involving connected wheels, such as gears or pulleys, the tangential velocity at the edge of one wheel equals that of another. This relationship is expressed as $r_1{\omega }_1=r_2{\omega }_2$. The angular speeds are inversely related to their radii, highlighting the importance of understanding rotational kinematics in mechanical systems. Mastery of these concepts aids in solving real-world engineering problems efficiently.

Intro to Connected Wheels

In problems involving multiple wheels, such as cylinders, discs, or gears, understanding the relationship between their rotational motions is crucial. These systems often consist of two or more wheels connected by a chain or belt, similar to a bicycle setup. The primary focus is on how these wheels interact when they rotate around a fixed axis.

When analyzing such systems, it is essential to recognize that the tangential velocity at the edge of one wheel is equal to the tangential velocity at the edge of the second wheel. This can be expressed mathematically as:

$$v_{t1} = v_{t2}$$

For a fixed axis, the tangential velocity \(v_t\) is related to the radius \(r\) and angular velocity \(\omega\) by the equation:

$$v_t = r \omega$$

Thus, for two connected wheels, we can derive the relationship:

$$r_1 \omega_1 = r_2 \omega_2$$

This equation highlights that while the tangential velocities are equal, the angular velocities \(\omega\) will differ if the radii \(r\) of the wheels are different. Specifically, a larger radius results in a smaller angular velocity, demonstrating an inverse relationship between radius and angular speed.

Additionally, there are various ways to express angular velocity, including frequency \(f\), period \(T\), and revolutions per minute (RPM). These relationships can be summarized as follows:

1. $$\omega = 2 \pi f$$

2. $$\omega = \frac{2 \pi}{T}$$

3. $$\omega = \frac{2 \pi \text{RPM}}{60}$$

By substituting these expressions into the original equation, we can derive alternative forms:

For frequency:

$$r_1 f_1 = r_2 f_2$$

For period:

$$\frac{r_1}{T_1} = \frac{r_2}{T_2}$$

For RPM:

$$r_1 \text{RPM}_1 = r_2 \text{RPM}_2$$

These variations provide flexibility in solving problems involving connected wheels. For example, if we have two gears with radii \(r_1 = 2\) and \(r_2 = 3\), and the smaller gear rotates at an angular speed of \(40\) radians per second, we can find the angular speed of the larger gear using the equation:

$$\omega_2 = \frac{r_1 \omega_1}{r_2}$$

Substituting the known values gives:

$$\omega_2 = \frac{2 \times 40}{3} = 26.7 \text{ radians per second}$$

This example illustrates the straightforward application of these principles in rotational kinematics, emphasizing the importance of understanding the relationships between radius, angular velocity, and tangential velocity in systems of connected wheels.

Speed of pulleys of different radii

In a system involving two pulleys with radii \( r_1 = 0.3 \, \text{m} \) and \( r_2 = 0.4 \, \text{m} \), a light cable runs through the edges of both pulleys. The relationship between the linear velocity of the cable and the angular velocities of the pulleys is described by the equation:

\( r_1 \omega_1 = r_2 \omega_2 \)

Here, \( \omega_1 \) and \( \omega_2 \) represent the angular velocities of the first and second pulleys, respectively. When the cable is pulled, it causes both pulleys to spin. The linear velocity \( v \) of the cable is constant throughout the system, meaning:

\( v_{\text{cable}} = v_1 = v_2 \

Given that the speed of the cable is \( v = 5 \, \text{m/s} \), we can express the angular velocities in terms of the radii:

\( v = r_1 \omega_1 \quad \text{and} \quad v = r_2 \omega_2 \

To find \( \omega_1 \), we rearrange the equation:

\( \omega_1 = \frac{v}{r_1} = \frac{5}{0.3} \approx 16.67 \, \text{radians/second} \

Similarly, for \( \omega_2 \):

\( \omega_2 = \frac{v}{r_2} = \frac{5}{0.4} = 12.5 \, \text{radians/second} \

Thus, the angular speeds of the pulleys are approximately \( 16.67 \, \text{radians/second} \) for the first pulley and \( 12.5 \, \text{radians/second} \) for the second pulley. A crucial takeaway from this scenario is that the linear velocity of the cable is equal to the tangential velocities at the edges of both pulleys, allowing for the application of the aforementioned relationships.

Here’s what students ask on this topic:

What is the relationship between the tangential velocities of two connected wheels?

The tangential velocities at the edges of two connected wheels are equal. This relationship is crucial in problems involving gears, pulleys, or any connected rotational systems. Mathematically, this is expressed as:

$v_{\mathrm{t1}}=v_{\mathrm{t2}}$

Since the tangential velocity $v$ is given by $v=r\omega$, where $r$ is the radius and $\omega$ is the angular velocity, the relationship can be written as:

$r_1{\omega }_1=r_2{\omega }_2$

How do you calculate the angular speed of a larger wheel connected to a smaller wheel?

To calculate the angular speed of a larger wheel connected to a smaller wheel, you use the relationship between their radii and angular speeds. The formula is:

$r_1{\omega }_1=r_2{\omega }_2$

Solving for the angular speed of the larger wheel ${\omega }_2$, we get:

${\omega }_2=\frac{r_1}{r_2}{\omega }_1$

For example, if the smaller wheel has a radius of 2 units and an angular speed of 40 radians per second, and the larger wheel has a radius of 3 units, the angular speed of the larger wheel is:

${\omega }_2=\frac{2}{3}*\; 40\; =\; 26.7$ radians per second.

What are the different ways to describe how quickly something spins?

There are four main ways to describe how quickly something spins:

1. Angular Velocity ($\omega$): Measured in radians per second (rad/s).
2. Frequency ($f$): Measured in Hertz (Hz), it represents the number of rotations per second.
3. Period ($T$): Measured in seconds (s), it is the time taken for one complete rotation. It is the inverse of frequency: $T=\frac{1}{f}$.
4. Revolutions Per Minute (RPM): It represents the number of rotations per minute. It is related to frequency by: $f=\frac{\mathrm{RPM}}{60}$.

How do you convert angular velocity to frequency and period?

To convert angular velocity ($\omega$) to frequency ($f$) and period ($T$), you can use the following relationships:

Frequency:

$\omega =\; 2\pi f$

Solving for $f$, we get:

$f=\frac{\omega }{2}$

Period:

The period is the inverse of frequency:

$T=\frac{1}{f}$

Combining the two equations, we get:

$T=\frac{2}{\pi }\frac{1}{\omega }$

What is the significance of the equation r1⁢ω1=r2⁢ω2 in connected wheels problems?

The equation $r_1{\omega }_1=r_2{\omega }_2$ is fundamental in problems involving connected wheels, such as gears or pulleys. It states that the product of the radius and angular velocity of one wheel is equal to that of the other wheel. This relationship ensures that the tangential velocities at the points where the wheels are connected are equal, which is crucial for the system to function correctly. Understanding this equation helps in solving real-world engineering problems efficiently, such as designing gear systems in machinery or understanding the mechanics of bicycles.