A bob of mass `m` hangs from the ceiling of a smooth trolley car which is moving with a constant acceleration a. If young's modulus, radius and length of the string are `Y, r` and `I`, respectively, find the (a) stress in the string and (b) extension of the string when it makes a constant angle relative to vertical.

Answer

`(m(sqrt(g^(2)+a^(2))l)/(pir^(2)Y))`

Explanation

Relative to the accelerating frame (trolley car), the forces acting on the bob are as shown in the figure.

Then `T=msqrt(a^(2)+g^(2))` ltbr. Since the string is light, same tension is felt at all points of the string.
Putting the area of cross section of the string `A=pir^(2)` in the formula
`sigma=(F')/a`
where `F'=T= msqrt(a^(2)+g^(2))`
`sigma=(msqrt(g^(2)+d^(2)))/(pir^(2))`
b. As `Y=(F//A)/(/_\l//l)=(sigmal)/(/_\l)`
which gives `/_\l=(sigmal)/Y`
Hence `/_\l=(m|veca-vecg|l)/(pir^(2))=(m(sqrt(g^(2)+a^(2)))l)/(pir^(2)Y)`