To simulate car accidents,auto manufacturers | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given: Mass $m = 1000 \; kg$,speed $v = 18.0 \; km/h = 18.0 	imes (5/18) \; m/s = 5 \; m/s$,coefficient of friction $\mu = 0.5$,spring constant $

Vedclass · Accepted Answer

$1.35$

Vedclass · Answer

(A) Given: Mass $m = 1000 \; kg$,speed $v = 18.0 \; km/h = 18.0 	imes (5/18) \; m/s = 5 \; m/s$,coefficient of friction $\mu = 0.5$,spring constant $k = 6.25 	imes 10^{3} \; N/m$,and $g = 10.0 \; m/s^2$.In the presence of friction,the work-energy theorem states that the change in kinetic energy is equal to the total work done by all forces (spring force and frictional force).$\Delta K = W_{spring} + W_{friction}$$0 - \frac{1}{2} m v^2 = -\frac{1}{2} k x_m^2 - \mu m g x_m$Rearranging gives the quadratic equation: $\frac{1}{2} k x_m^2 + \mu m g x_m - \frac{1}{2} m v^2 = 0$$k x_m^2 + 2 \mu m g x_m - m v^2 = 0$Substituting the values: $(6.25 	imes 10^3) x_m^2 + 2(0.5)(1000)(10) x_m - (1000)(5^2) = 0$$6250 x_m^2 + 10000 x_m - 25000 = 0$Dividing by $1250$: $5 x_m^2 + 8 x_m - 20 = 0$Using the quadratic formula $x_m = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$:$x_m = \frac{-8 + \sqrt{64 - 4(5)(-20)}}{2(5)} = \frac{-8 + \sqrt{64 + 400}}{10} = \frac{-8 + \sqrt{464}}{10} \approx \frac{-8 + 21.54}{10} = 1.354 \; m$.Thus,the maximum compression is approximately $1.35 \; m$.

Similar Questions

$A$ lift of mass $2000 \ kg$ starts from rest in the basement and moves to the fourth floor,which is at a height of $25 \ m$. As it passes the fourth floor,its speed is $3 \ ms^{-1}$. There is a constant frictional force of $500 \ N$ acting on it. Calculate the work done by the lift's motor in $kJ$.

Water falls from a height of $60 \, m$ at the rate of $15 \, kg/s$ to operate a turbine. The losses due to frictional force are $10 \%$ of the input energy. How much power is generated by the turbine? $(g=10 \, m/s^2)$ (In $kW$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module