Question 24: Work, Energy and Power & Circular Motion

Question

A car moving with a speed of $54 \mathrm{~km} / \mathrm{h}$ takes a turn of radius 20 m . A simple pendulum is suspended from the ceiling of the car. Determine the angle made by the string of the pendulum with the vertical during the turning. (Take $g=10 \mathrm{~m} / \mathrm{s}^2$ )

Accepted Answer

The speed of the car is given in $\mathrm{km} / \mathrm{h}$. We must convert it to $\mathrm{m} / \mathrm{s}$ to match the units of gravity and radius.
$$ \mathrm{v}=54 	imes \frac{5}{18}=3 	imes 5=15 \mathrm{~m} / \mathrm{s} $$
When the car takes a turn, from the perspective of an observer inside the car, the pendulum bob experiences gravity ( mg ) which is acting vertically downwards and centrifugal force ( $\mathrm{F}_{\mathrm{c}}$ ) a pseudo-force acting horizontally outwards due to the circular motion.
$$ \mathrm{F}_{\mathrm{c}}=\frac{\mathrm{mv}^2}{\mathrm{r}} $$
Where m is mass, v is velocity, and r is radius.
Let the string make an angle $	heta$ with the vertical. The tension T in the string can be resolved into two components:
The vertical component is $\mathrm{T} \cos 	heta$
The horizontal component is $\mathrm{T} \sin 	heta$
In equilibrium (relative to the car) :
$$\mathrm{T} \cos 	heta=\mathrm{mg} 	ag{i}$$ .....(i)
$$\mathrm{T} \sin 	heta=\frac{\mathrm{mv}^2}{\mathrm{r}} \ldots (ii)$$
Dividing equation (ii) by the equation (i) :
$$ \frac{\mathrm{T} \sin 	heta}{\mathrm{~T} \cos 	heta}=\frac{\mathrm{mv}^2 / \mathrm{r}}{\mathrm{mg}} $$
$$\Rightarrow $$ $$ 	an 	heta=\frac{\mathrm{v}^2}{\mathrm{rg}} $$
Substituting the given values, where speed of car is $\mathrm{v}=15 \mathrm{~m} / \mathrm{s}$, radius of turn is $\mathrm{r}=20 \mathrm{~m}$ and the acceleration due to gravity is $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$
$$ 	an 	heta=\frac{15^2}{20 	imes 10} $$
$$\Rightarrow $$ $	an 	heta=\frac{9}{8}=1.125$
$$\Rightarrow $$ $$ 	heta=	an ^{-1}(1.125) $$
Therefore, the angle made by the string with the vertical is $	an ^{-1}(1.125)$.
Hence, the correct option is (C).

Answer

$$ 	an ^{-1}(0.5) $$

Answer

$$ 	an ^{-1}(0.75) $$

Answer

$$ 	an ^{-1}(1.125) $$

Answer

$$ 	an ^{-1}(0.25) $$

Work, Energy and Power & Circular Motion

A car moving with a speed of 54 \mathrm{~km} / \mathrm{h} takes a turn of radius 20 m . A simple pendulum is suspended from the ceiling of the car. Determine the angle made by the string of the pendulum with the vertical during the turning. (Take g=10 \mathrm{~m} / \mathrm{s}^2 )

Choose the correct answer: