Question

In a bicycle the radius of rear wheel is twice the radius of front wheel. If \(\mathrm{r}_{\mathrm{F}}\) and \(\mathrm{r}_{\mathrm{r}}\) are the radius, \(\mathrm{v}_{\mathrm{F}}\) and \(\mathrm{v}_{\mathrm{r}}\) are speed of top most points of wheel respectively then... \(\\{\mathrm{A}\\} \mathrm{v}_{\mathrm{r}}=2 \mathrm{v}_{\mathrm{F}}\) \(\\{\mathrm{B}\\} \mathrm{v}_{\mathrm{F}}=2 \mathrm{v}_{\mathrm{r}} \quad\\{\mathrm{C}\\} \mathrm{v}_{\mathrm{F}}=\mathrm{v}_{\mathrm{r}}\) \(\\{\mathrm{D}\\} \mathrm{v}_{\mathrm{F}}>\mathrm{v}_{\mathrm{r}}\)

Short answer

The correct answer is option A: \(v_r = 2v_F\).

Step-by-step solution

Write down the given information

We are given the following: 1. The radius of the rear wheel is twice the radius of the front wheel: \(r_r = 2r_F\) 2. The speeds of the top points of the wheels are \(v_F\) and \(v_r\).

Find the relationship between the linear and angular velocities of the two wheels

To find the relationship between the wheels' speeds, we will use the formula connecting linear and angular velocity: \(v = r\omega\), where \(v\) is the linear velocity, \(r\) is the radius, and \(\omega\) is the angular velocity. For the front wheel, we have: \[v_F = r_F \omega_F\] For the rear wheel, we have: \[v_r = r_r \omega_r\]

Use the given information to relate the angular velocities

Since the radius of the rear wheel is twice the radius of the front wheel, we can create an expression for the angular velocities in terms of linear velocities: \[\omega_F = \frac{v_F}{r_F}\] and \[\omega_r = \frac{v_r}{r_r} = \frac{v_r}{2r_F}\] Now, note that both the wheels are attached to the same bicycle, which means they will always be in contact with the ground. Hence, their angular velocities will be equal: \[\omega_F = \omega_r\]

Solve for the relationship between the linear velocities

Substitute the expressions for \(\omega_F\) and \(\omega_r\) from Step 3 into the equation \(\omega_F = \omega_r\): \[\frac{v_F}{r_F} = \frac{v_r}{2r_F}\] Now, solve for the relationship between \(v_F\) and \(v_r\): \(v_F = \frac{1}{2}v_r\) Which is equivalent to: \(v_r = 2v_F\)

Identify the correct answer

Comparing our result with the given options, we find that our result matches option A: \(v_r = 2v_F\) Hence, the correct answer is option A.

Key concepts

Angular Velocity

Angular velocity describes how fast something spins or rotates. It’s a measure of the angle an object moves through in a certain time. Imagine a spinning top. The angular velocity tells us how quickly the top is spinning around its axis.
For wheels, angular velocity (\( \omega \)) connects directly with linear velocity (\( v \)) through the equation \( v = r\omega \), where \( r \) is the radius. This equation indicates that for a given angular velocity, the linear velocity increases with a larger radius.
In our bicycle problem, both wheels have the same angular velocity because they are part of the same bike. Whatever speed the cyclist is pedaling at, it gets transferred equally to both wheels in terms of angular velocity. Thus, understanding angular velocity helps us understand how wheels of different shapes or sizes interact on a bicycle.

Linear Velocity

Linear velocity refers to how fast something moves in a straight line. It is the distance traveled per unit of time, like how fast a car drives down the road.
For rotating objects like wheels, linear velocity relates to angular velocity. Remember that formula \( v = r\omega \)? This shows linear velocity (\( v \)) depends on the product of the radius (\( r \)) and angular velocity (\( \omega \)).
In our exercise, the linear velocity at the top of the rear wheel is influenced by the fact that it has a larger radius than the front wheel. This results in a higher linear speed for the rear wheel, even though both wheels rotate with the same angular velocity, leading to the answer \( \mathrm{v}_{\mathrm{r}} = 2\mathrm{v}_{\mathrm{F}} \).

Radius Ratio

The radius of a circle is the distance from its center to its edge. Ratio means a comparative value between two quantities. So, the radius ratio is simply comparing the sizes of two circles.In this case, the rear wheel’s radius is twice as large as the front wheel’s radius. Mathematically, \( r_r = 2r_F \). This ratio impacts how the linear velocities are calculated. Since the angular velocities are the same, doubling the radius also doubles the linear velocity due to \( v = r\omega \).This radius ratio is critical in understanding why the rear wheel’s speed is twice that of the front wheel, given their angular velocities are equal. Thus, the mechanical advantage of having wheels of different sizes can lead to different speeds.

Rotational Motion

Rotational motion occurs when an object turns around a center or an axis. It’s a type of motion that’s fundamental in many mechanical systems, especially those involving wheels. Think about how a bicycle moves. The wheels rotate on their axes, propelling the bike forward. This type of motion involves both linear and angular velocity, where linear velocity helps the bike move forward, and angular velocity keeps the wheels turning. Understanding rotational motion in bicycles is essential for solving problems involving different wheel sizes. Despite the rear and front wheels having the same rate of angular rotation, their rotational motion leads to different linear speeds due to the difference in radii. It showcases how bicycles achieve motion efficiency through rotational dynamics.