A heavy ball of mass M is suspended from the ceiling of a car by a light string of mass m(m << M). The speed of transverse waves in the string is 50 ms -1 when the car is at rest. But if the car is imparted an acceleration of a, the wave speed increases to 51 `ms^-1`. What is the value of a? [Take `g = 10 ms^-2]`

Answer

`02.82`

Explanation

Case I:

From diagram, `T_1=mg`
`:. Upsilon_1=sqrt(T_1/mu)=sqrt(mg)/mu" "…(i)`
Case II:

From diagram,
`T_2 cos theta=mg, T_2 sin theta=ma`
`:. T_2^2(sin^2theta+cos^2 tehta)`
`=m^2(g^2+a^2)`
`T_2=m(g^2+a^2)^(1/2)`
`upsilon_2=sqrt(T_2/mu)=sqrt((msqrt(a^2+g^2))/mu)" "...(ii)`
From (i) and (ii), `upsilon_1/upsilon_2=sqrt(g/(g^2+a^2)^(1/2))`
`impliesv_2/v_1=1+1/4 a_2/g_2" "...`
`[:.(1+X)^n~~1+nx]`
`51/50=1+1/4 a_2/g_2`
`impliesa^2/g^2=8/100`
`impliesa=(2sqrt2""g)/10`