First slide

Torque

Question

An automobile moves on a road with a speed of 54 km $h^{- 1}$ . The radius of its wheels is 0.45 m and the moment of inertia of the wheel about its axis of rotation is 3 $k g m^{2}$ . If the vehicle is brought to rest in 15 s, the magnitude of average torque transmitted by its brakes to the wheel is

Moderate

A

$10.86 kg m^{2} s^{- 2}$
B

$2.86 kg m^{2} s^{- 2}$
C

$6.66 kg m^{2} s^{- 2}$
D

$8.58 kg m^{2} s^{- 2}$

Solution

Here,
speed of the automobiles,
$v = 54 km h^{- 1} = 54 \times \frac{5}{18} m s^{- 1} = 15 m s^{- 1}$ .
Radius of the wheel of the automobile, R=0.45 m Moment of inertia of the wheel about its axis of rotation, $I = 3 kg m^{2}$
Time in which the vehicle brought to rest, t = 15 s. The initial angular speed of the wheel is
$ω_{i} = \frac{v}{R} = \frac{15 m s^{- 1}}{0.45 m} = \frac{1500}{45} {rads}^{- 1} = \frac{100}{3} {rads}^{- 1}$
and its final angular speed is $ω_{f} = 0 (as the vehicle comes to rest)$
$∴ The angular retardation of the wheel is$
$α = \frac{ω_{f} - ω_{i}}{t} = \frac{0 - \frac{100}{3}}{15 s} = - \frac{100}{45} rad s^{- 2}$
$The magnitude of required torque is$
$τ = I | α | = (3 kg m^{2}) (\frac{100}{45} {rads}^{- 2})$
$= \frac{20}{3} kg m^{2} s^{- 2} = 6.66 kg m^{2} s^{- 2}$

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