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.