A car starts from rest on a horizontal circular road of radius 190 m and gains speed at a uniform rate `1*2 m//s^(2)` . The coefficient of static between the road and the tyres is `0*37` . Calculate the distance travelled by the car before it begins to skid.

Explanation

When car is moving in non- uniform circular motion , then `F_(1) and F_(2) ` both types of forces are acting
` F_("net")= sqrt(F_(c)^(2)+ F_(t)^(2))`
` = sqrt (m^(2)a_(c)^(2) +m^(2)a_(t)^(2))`
` = sqrt(a_(c)^(2) + a_(t)^(2))`
` = m sqrt(v^(4)/R^(2)+a_(t)^(2))`
Maximum force of friction between car and road is
`f_(max) = mu_(s)N`
` = mu_(s)mg`
`= F_("net")`
` mu_(s)mg = msqrt (v^(2)/R^(2) +a_(t)^(2))`
Squaring on both side
` mu_(s)^(2)g^(2) = v^(4)/R^(2) + a_(t)^(2)`
` rArr (v^(4)/R^(2)) = (mu_(s)g)^(2) - a_(t)^(2)`
` v^(4) = R^(2) [ mu_(s)^(2)g^(2)- a_(t)^(2)]`
` v_(max) = (R^(2)[mu_(s)^(2) - g^(2)- a_(t)^(2)])^(1//4)`
Distance travelled before skidding , `S = (v_(max)^(2))/(2a)`
` S = (R(mu_(s)^(2)g^(2) - a_(t)^(2))^(1//2))/(2a_(t))`= 270 m `