Question

A cyclist turns around a curve at 15 miles/hour. If he turns at double the speed the tendency to overturn is
A. Double
B. Quadrupled
C. Halved
D. Unchanged

Answer 1

The tendency to overturn depends on the centripetal force acting towards the centre.
The centripetal force depends on the square of velocity. Using this we can find the effect of doubling the velocity in the tendency to overturn.

Complete step by step answer:
It is given that a cyclist turns around a curve at $15\,{\text{miles/hour}}$.
$\Rightarrow v = 15\,{\text{miles/hour}}$
We need to find how the tendency to overturn changes when he turns at double the speed.
We know that in a circular motion there is centripetal force acting on the cyclist.
It is the centripetal force that helps him to turn.
The centripetal force is calculated as
${F_c} = \dfrac{{m{v^2}}}{r}$
Where, m is the mass, v is the velocity, r is the radius of the circular path.
Let the initial velocity be denoted as ${v_i}$ .
Then the centripetal force will be
$\Rightarrow {F_c} = \dfrac{{m{v_i}^2}}{r}$ ………………….(1)
We need to find the tendency to overturn when ${v_i}$ becomes $2{v_i}$ .
When the velocity is $2{v_i}$, the centripetal force will be
$\Rightarrow {F_c}^\prime = \dfrac{{m{v_f}^2}}{r}$
$\Rightarrow {F_c}^\prime = \dfrac{{m{{\left( {2{v_i}} \right)}^2}}}{r}$
$\Rightarrow {F_c}^\prime = \dfrac{{4m{v_i}^2}}{r}$ ………...(2)
On comparing the equations 1 and 2 we can see that the tendency to overturn has changed.
Let us divide equation two with one, then we get
$\Rightarrow \dfrac{{{F_c}^\prime }}{{{F_c}}} = \dfrac{{\dfrac{{4m{v_i}^2}}{r}}}{{\dfrac{{m{v_i}^2}}{r}}}$
$\Rightarrow \dfrac{{{F_c}^\prime }}{{{F_c}}} = 4$
$\Rightarrow {F_c}^\prime = 4{F_c}$
We can see that the centripetal force became four times the initial value. Which means, the tendency to overturn has increased four times.
Hence, we can say that the tendency to overturn is quadrupled.

Therefore, the correct answer is option B.

Note:
There are two forces connected with circular motion- centripetal force and centrifugal force. Don't get confused between these two forces. Centripetal force is a force that is always acting towards the centre of the circular path. Whereas, centrifugal force is a force that acts away from the centre.