Question 190: Waves

Question

A heavy ball of mass M is suspendeed from the ceiling of a car by a light string of mass m (m < < M). When the car is at rest, the speed of transverse waves in the string is 60 ms$$-$$1. When the car has acceleration a, the wave-speed increases to 60.5 ms$$-$$1. The value of a, in terms of gravitational acceleration g, is closest to :

Accepted Answer

Resultant force on the ball of mass M when car is moving with a acceleration
a
is ,
F
net
= $$\sqrt {{{\left( {Mg} ight)}^2} + {{\left( {Ma} ight)}^2}} $$
= $$M\sqrt {{g^2} + {a^2}} $$
$$ 	herefore $$   T = M$$\sqrt {{g^2} + {a^2}} $$
We know,
Velocity, V = $$\sqrt {{T \over \mu }} $$
When Car is at rest then,
60 = $$\sqrt {{{Mg} \over \mu }} $$   . . . . (1)
and when is moving then
60.5 =  $$\sqrt {{{M\sqrt {{g^2} + {a^2}} } \over \mu }} $$    . . . . (2)
By dividing (2) by (1) we get,
$${{60.5} \over {60}} = \sqrt {{{\sqrt {{g^2} + {a^2}} } \over g}} $$
$$ \Rightarrow $$   $$\left( {1 + {{0.5} \over {60}}} ight)$$ = $${\left( {{{{g^2} + {a^2}} \over {{g^2}}}} ight)^{{1 \over 4}}}$$
$$ \Rightarrow $$   $${{{g^2} + {a^2}} \over {{g^2}}}$$ = $${\left( {1 + {{0.5} \over {60}}} ight)^4}$$
$$ \Rightarrow $$   $${{{g^2} + {a^2}} \over {{g^2}}}$$ = 1 + 4 $$ 	imes $$ $${{{0.5} \over {60}}}$$ [Using Binomial approximation]
$$ \Rightarrow $$   $${{{g^2} + {a^2}} \over {{g^2}}}$$ = 1 + $${1 \over {30}}$$
$$ \Rightarrow $$   1 + $${{{a^2}} \over {{g^2}}}$$ = 1 + $${1 \over {30}}$$
$$ \Rightarrow $$   $${a \over g}$$ = $${1 \over {\sqrt {30} }}$$
$$ \Rightarrow $$   a = $${g \over {\sqrt {30} }}$$
$$ 	herefore $$   Closest answer, a = $${g \over 5}$$

Answer

$${g \over {30}}$$

Answer

$${g \over 5}$$

Answer

$${g \over 10}$$

Answer

$${g \over 20}$$

Waves

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