Question

A small car of mass m and a large car of mass 4m drive along a highway at constant speed. They approach a curve of radius R. Both cars maintain the same acceleration a as they travel around the curve. How does the speed of the small car vS compare to the speed of the large car vL as they round the curve?

A small car of mass m and a large car of mass 4m drive along a highway at constant speed. They approach a curve of radius R. Both cars maintain the same acceleration a as they travel around the curve. How does the speed of the small car vS compare to the speed of the large car vL as they round the curve? Now assume that two identical cars of mass m drive along a highway. One car approaches a curve of radius 2R at speed v. The second car approaches a curve of radius 6R at a speed of 3v. How does the magnitude F1 of the net force exerted on the first car compare to the magnitude F2 of the net force exerted on the second car?

Now assume that two identical cars of mass m drive along a highway. One car approaches a curve of radius 2R at speed v. The second car approaches a curve of radius 6R at a speed of 3v. How does the magnitude F1 of the net force exerted on the first car compare to the magnitude F2 of the net force exerted on the second car?

Answer 1

Concepts and reason

The concepts used to solve this problem are circular motion for an object and centripetal force for a circular motion.

Initially, the centripetal force can be calculated for the small car which is moving in a circular motion. Later, the centripetal force can be calculated for the large car which is also moving in a circular motion.

Finally, both the force has to equate to compare the velocity for the both cars.

Initially, the centripetal force can be calculated for the both cars which are moving in a circular motion and also having same mass.

Finally, both the force has to be equated to compare the force acting on the cars.

Fundamentals

The centripetal acceleration for the small car is,

${a_s} = \frac{{v_s^2}}{R}$

Here, ${a_s}$ is the centripetal acceleration of the small car, ${v_s}$ is the velocity of the small car, and $R$ is the radius of round curve.

The centripetal acceleration for the large car is,

${a_L} = \frac{{v_L^2}}{R}$

Here, ${a_L}$ is the centripetal acceleration of the large car and ${v_L}$ is the velocity of the large car.

The centripetal force for the first car is,

${F_1} = \frac{{mv_1^2}}{{{R_1}}}$

Here, ${F_1}$ is the centripetal force for the first car, ${v_1}$ is the velocity of the first car, $m$ is the mass of the first car and ${R_1}$ is the radius of curvature.

The centripetal force for the second car is,

${F_2} = \frac{{mv_2^2}}{{{R_2}}}$

Here, ${F_2}$ is the centripetal force for the large car, ${v_2}$ is the velocity of the second car, $m$ is the mass of the second car and ${R_2}$ is the radius of curvature.

The expression for the centripetal acceleration for the small car is,

${a_s} = \frac{{v_s^2}}{R}$

The expression for the centripetal acceleration for the large car is,

${a_L} = \frac{{v_L^2}}{R}$

Both the cars maintain the same acceleration. So, equate the acceleration of the small and large car.

Equate the expression ${a_s}$ and ${a_L}$ and simplify.

$\begin{array}{l}\\\frac{{v_s^2}}{R} = \frac{{v_L^2}}{R}\\\\v_s^2 = v_L^2\\\\{v_s} = {v_L}\\\end{array}$

The expression for the centripetal force for the first car is,

${F_1} = \frac{{mv_1^2}}{{{R_1}}}$

Substitute, $v$ for ${v_1}$ and $\left( {2R} \right)$ for ${R_1}$

${F_1} = \frac{{m{v^2}}}{{2R}}$

The expression for the centripetal force for the second car is,

${F_2} = \frac{{mv_2^2}}{{{R_2}}}$

Substitute, $\left( {3v} \right)$ for ${v_2}$ and $\left( {6R} \right)$ for ${R_2}$

$\begin{array}{l}\\{F_2} = \frac{{m{{\left( {3v} \right)}^2}}}{{6R}}\\\\{F_2} = \frac{{9m{v^2}}}{{6R}}\\\end{array}$

Take the ratio of the forces ${F_1}$ and ${F_2}$ .

$\begin{array}{l}\\\frac{{{F_1}}}{{{F_2}}} = \frac{{\left( {\frac{{m{v^2}}}{{2R}}} \right)}}{{\left( {\frac{{9m{v^2}}}{{6R}}} \right)}}\\\\\frac{{{F_1}}}{{{F_2}}} = \frac{1}{3}\\\\{F_1} = \frac{1}{3}{F_2}\\\end{array}$

Ans:

The speed of the small car relative to the large car is ${v_s} = {v_L}$ .

The force exerted on first car compared to the second car is ${F_1} = \frac{1}{3}{F_2}$