More Connect Wheels (Bicycles): Videos & Practice Problems

Bicycle problems involve understanding the relationships between various components like pedals, sprockets, and wheels. Key equations include $v=r\omega$, where v is linear speed, r is radius, and ω is angular speed. For moving bikes, the front and back wheels typically share the same radius, leading to equal angular velocities. Understanding these dynamics is crucial for solving problems related to bike motion and mechanics.

Bicycle Problems (Static)

Bicycle problems involve understanding the relationships between various components of a bike, which typically include five main parts: pedals, middle sprocket, back sprocket, back wheel, and front wheel. The complexity arises from the interconnectedness of these parts, where the motion of one affects the others.

Starting with the pedals, they initiate the motion by causing the middle sprocket to spin. This relationship can be expressed as:

\( \omega_1 = \omega_2 \)

where \( \omega_1 \) is the angular velocity of the pedals and \( \omega_2 \) is the angular velocity of the middle sprocket. Both components rotate around the same axis, meaning they share the same angular velocity.

The middle sprocket is connected to the back sprocket via a chain, which ensures that the tangential velocities at the edges of both sprockets are equal. This relationship can be described by the equation:

\( r_2 \omega_2 = r_3 \omega_3 \)

Here, \( r_2 \) and \( r_3 \) are the radii of the middle and back sprockets, respectively, while \( \omega_3 \) is the angular velocity of the back sprocket.

Next, the back sprocket is connected to the back wheel, which leads to another relationship:

\( \omega_3 = \omega_4 \

where \( \omega_4 \) is the angular velocity of the back wheel. This means that the back sprocket and back wheel also share the same angular velocity.

In scenarios where the bike is static, such as when it is lifted off the ground, the front wheel does not spin. The motion of the pedals only affects the back sprocket and back wheel, as the front wheel relies on the bike's movement to rotate.

To illustrate these concepts, consider a problem where the diameters of the middle and back sprockets are given as 16 cm and 10 cm, respectively, and the pedals are spun at 8 radians per second. The angular velocities can be calculated as follows:

1. For the middle sprocket:

\( \omega_2 = \omega_1 = 8 \, \text{radians/second} \)

2. For the back sprocket, using the relationship:

\( r_2 \omega_2 = r_3 \omega_3 \)

Convert diameters to radii: \( r_2 = 0.08 \, \text{m} \) and \( r_3 = 0.05 \, \text{m} \). Thus:

\( 0.08 \times 8 = 0.05 \omega_3 \)

Solving for \( \omega_3 \) gives:

\( \omega_3 = \frac{0.08 \times 8}{0.05} = 12.8 \, \text{radians/second} \)

3. For the back wheel:

\( \omega_4 = \omega_3 = 12.8 \, \text{radians/second} \)

4. Finally, the front wheel does not spin, so:

\( \omega_5 = 0 \, \text{radians/second} \)

Understanding these relationships and calculations is crucial for solving bicycle-related problems in physics, as they illustrate the principles of rotational motion and the conservation of angular velocity across connected components.

RPM of pedals of static bicyle

When analyzing the relationship between the different components of a bicycle's drivetrain, it's essential to understand how the rotation of the pedals, sprockets, and wheels interconnect. In this scenario, we have a back sprocket with a diameter \(d\) and a middle sprocket with a diameter \(2d\). The relationship between the diameters can be converted to radii, where the radius of the back sprocket is \(r\) and the radius of the middle sprocket is \(2r\).

To determine the revolutions per minute (RPM) of the pedals based on the RPM of the back wheel, we can establish several key relationships. The first relationship is between the pedals and the middle sprocket, which rotate at the same angular velocity, denoted as \(\omega\). This means that the RPM of the pedals (\(RPM_1\)) equals the RPM of the middle sprocket (\(RPM_2\)).

The second relationship involves the middle sprocket and the back sprocket, which share the same tangential velocity. This can be expressed mathematically as:

\[ r_2 \cdot \omega_2 = r_3 \cdot \omega_3 \]

Substituting the angular velocity with RPM gives:

\[ r_2 \cdot RPM_2 = r_3 \cdot RPM_3 \]

Finally, the third relationship is between the back sprocket and the back wheel, which also rotate at the same angular velocity, leading to:

\[ RPM_3 = RPM_4 \]

Given that the RPM of the back wheel (\(RPM_4\)) is \(x\), we can deduce that \(RPM_3\) is also \(x\). To find \(RPM_1\) in terms of \(x\), we can rearrange the equation from the second relationship:

\[ RPM_1 = \frac{r_3 \cdot x}{r_2} \]

Substituting \(r_3 = r\) and \(r_2 = 2r\) into the equation results in:

\[ RPM_1 = \frac{r \cdot x}{2r} = \frac{x}{2} \]

This indicates that the pedals will spin at half the RPM of the back wheel. This linear relationship highlights an important principle: if one wheel has double the radius of another, it will rotate at half the speed. Thus, as the size of the sprocket increases, its rotational speed decreases proportionally.

Understanding these relationships is crucial for effectively analyzing mechanical systems and predicting the behavior of interconnected components in a bicycle's drivetrain.

Bicycle Problems (Moving)

In analyzing bicycle motion, it's essential to understand the dynamics of the wheels when they are free to move. When a bike is stationary, the wheels do not contribute to any lateral movement, resulting in a fixed axis scenario where the velocity of the center of mass (VCM) of the wheel is zero. In this case, the angular velocity (ω) of the wheels is also zero unless the back wheel is spun independently, such as when the bike is lifted.

Conversely, when the bike is in motion, it operates under a free axis condition. Here, both the angular velocity (ω) and the linear velocity (v) are present. The relationship between these velocities can be expressed as:

$$ V_{CM} = r \cdot \omega $$

where r is the radius of the wheel. For most bicycles, the front and back wheels have the same radius, leading to the conclusion that:

$$ \omega_{front} = \omega_{back} $$

This means that both wheels rotate at the same angular speed when the bike is moving. The relationship between the pedal sprockets and the wheels can be summarized as follows:

1. The angular velocity of the pedal sprocket (ω₁) equals that of the rear sprocket (ω₂).

2. The angular velocity of the rear sprocket (ω₃) equals that of the back wheel (ω₄).

3. The linear velocities of the rear sprocket and back wheel are equal, leading to:

$$ r_2 \cdot \omega_2 = r_3 \cdot \omega_3 $$

4. The relationship between the back wheel and front wheel can be expressed as:

$$ r_4 \cdot \omega_4 = r_5 \cdot \omega_5 $$

When both wheels are in motion, their linear speeds at the center of mass are equal to the bike's speed. For example, if a bike travels at 15 m/s, both wheels also have a linear speed of 15 m/s. To find the angular speed of the wheels, the formula can be rearranged:

$$ \omega = \frac{v_{CM}}{r} $$

For instance, if the radius of the wheels is 0.66 m, the angular speed (ω) can be calculated as:

$$ \omega = \frac{15 \, \text{m/s}}{0.66 \, \text{m}} \approx 22.73 \, \text{radians/second} $$

This calculation applies equally to both wheels, confirming that they rotate at the same angular speed when the bike is in motion. Understanding these relationships is crucial for solving problems related to bicycle dynamics and motion.

Angular speeds of moving bicyle

In this scenario, we analyze the motion of a bicycle with two wheels, both having a radius of 0.70 meters. The bicycle's middle sprocket has a radius of 0.15 meters, while the back sprocket has a radius of 0.08 meters. When the bicycle is moving at a linear velocity of 20 meters per second, we can determine the angular speeds of the wheels and sprockets using the relationship between linear velocity and angular velocity.

The angular velocity, denoted as \( \omega \), can be calculated using the formula:

\( V_{cm} = r \cdot \omega \)

where \( V_{cm} \) is the center of mass velocity, \( r \) is the radius, and \( \omega \) is the angular speed.

For the front wheel (wheel 5), substituting the known values gives:

\( 20 = 0.70 \cdot \omega_5 \)

Solving for \( \omega_5 \) results in:

\( \omega_5 = \frac{20}{0.70} \approx 28.6 \, \text{radians per second} \)

Since the back wheel (wheel 4) has the same radius and is also moving at the same linear velocity, its angular speed \( \omega_4 \) is also 28.6 radians per second.

Next, we consider the back sprocket (sprocket 3). The angular speed of the back sprocket is the same as that of the back wheel, thus:

\( \omega_3 = 28.6 \, \text{radians per second} \)

To find the angular speed of the middle sprocket (sprocket 2), we use the relationship between the angular speeds of connected sprockets:

\( r_2 \cdot \omega_2 = r_3 \cdot \omega_3 \)

Rearranging this gives:

\( \omega_2 = \frac{r_3 \cdot \omega_3}{r_2} \)

Substituting the known values:

\( \omega_2 = \frac{0.08 \cdot 28.6}{0.15} \approx 15.3 \, \text{radians per second} \)

In summary, the angular speeds calculated are as follows: the front wheel and back wheel both have an angular speed of 28.6 radians per second, the back sprocket also shares this angular speed, while the middle sprocket has an angular speed of approximately 15.3 radians per second. This analysis illustrates the interconnectedness of the components in a bicycle's drivetrain and how linear motion translates into angular motion.

Here’s what students ask on this topic:

What are the key components involved in bicycle mechanics?

The key components in bicycle mechanics include the pedals, middle sprocket, back sprocket, back wheel, and front wheel. The pedals cause the middle sprocket to spin, which in turn drives the back sprocket via a chain. The back sprocket then causes the back wheel to spin. The front wheel spins only when the bike is in motion and touching the ground. Understanding the relationships between these components is crucial for solving problems related to bike motion and mechanics.

How do you calculate the angular velocity of a bicycle's back wheel?

To calculate the angular velocity (ω) of a bicycle's back wheel, you can use the relationship between the middle sprocket and the back sprocket. The equation is given by:

$\frac{r{2}_{}\omega 2_{}}{r{3}_{}}$

where r₂ and r₃ are the radii of the middle and back sprockets, respectively, and ω₂ is the angular velocity of the middle sprocket. By rearranging the equation, you can solve for ω₃:

$\omega 3_{}=\frac{r{2}_{}\omega 2_{}}{r{3}_{}}$

What happens to the front wheel of a bicycle when it is lifted off the ground?

When a bicycle is lifted off the ground, the front wheel does not spin because it is not in contact with the ground. The pedals, middle sprocket, back sprocket, and back wheel can still spin if you rotate the pedals, but the front wheel remains stationary. This is because there is no direct mechanical connection between the front wheel and the pedals or sprockets. The front wheel only spins when the bike is in motion and the wheels are touching the ground.

How do you determine the linear speed of a bicycle's wheels?

The linear speed (v) of a bicycle's wheels can be determined using the equation:

$v=r\omega$

where r is the radius of the wheel and ω is the angular speed. For a moving bicycle, the linear speed of the center of mass of both the front and back wheels is the same as the speed of the bike. If the bike moves at a speed of 15 m/s, then the linear speed of the center of mass of both wheels is also 15 m/s.

What is the relationship between the angular velocities of the front and back wheels of a moving bicycle?

For a moving bicycle, the angular velocities (ω) of the front and back wheels are typically the same if the wheels have the same radius. The relationship can be expressed as:

$r{4}_{}\omega 4_{}=r{5}_{}\omega 5_{}$

If the radii (r₄ and r₅) are equal, then the angular velocities (ω₄ and ω₅) are also equal. This ensures that both wheels move together without changing the distance between them.