Question

2.A car going around a certain curve at a speed of 25 km/h has
centripetal force acting on it of 100 N. If the speed of the car is
doubled, the centripetal force:​

Answer 1

$\bigstar$ Explanation $\bigstar$

$\leadsto$ Solution:-

We know that Centripetal acceleration = $\rm \frac{v^2}{r}$, where v is the magnitude of the linear velocity and r is the radius.

Therefore,

Centripetal force = Centripetal acceleration $\times$ Mass of the particle

Centripetal force = $\rm m \times \frac{v^2}{r} = \frac{mv^2}{r}$

Let us assume that the initial speed as $\rm V_o$,

Here in this question, the radius of the circular path is not changing, therefore, the initial radius = the final radius = r

Also, the mass of the particle is not mass so let's consider the mass of the particle as m.

Let the initial centripetal force be $\rm F_c$

$\rm F_c = \frac{mV_o^2}{r} --- eqn (i)$

According to the question the final speed = 2 $\times$ initial speed = $\rm 2V_o$

Let the final centripetal force be $\rm f_c$

$\rm f_c = \frac{m \times (2V_o)}{r} = \frac{4mV_o^2}{r}$

From eqn(i)

$\rm f_c = 4F_c$

$\leadsto$ Derivation of Centripetal Acceleration:-

Let's assume that the circular motion performed by the particle is uniform circular motion where the speed of the particle is constant and also the $\omega$ (Angular velocity) is constant.

We know that $\rm v = r\omega$, where v is the magnitude of the linear velocity and r is the radius

Now let's differentiate both sides with respect to time

$\rm \dfrac{dv}{dt} = \dfrac{d(r\omega)}{dt}$

According to product rule which says that,

If f(x) = g(x)h(x)

Then $\rm \dfrac{d[f(x)]}{dx} = \dfrac{d[g(x)]}{dx}h(x) + \dfrac{d[h(x)]}{dx}g(x)$

Therefore,

$\rm \dfrac{dv}{dt} = \dfrac{dr}{dt}\omega + \dfrac{d\omega}{dt}r$

As $\omega$ is constant, therefore $\frac{d\omega}{dt} =0$

$\rm \dfrac{dv}{dt} = \dfrac{dr}{dt}\omega$

We know that $\rm \frac{dr}{dt} = v$, also $\rm \frac{dv}{dt} = a_c$

$\rm a_c = v\omega$

We know that v = r$\omega$

Therefore,

$\rm a_c = r\omega^2$

From the formula v = r$\omega$ ⇒ $\rm \omega = \frac{v}{r}$

$\rm a_c = r \times \frac{v^2}{r^2} = \frac{v^2}{r}$