Question

What is the maximum speed with which a 1200 kg car can round a turn of radius 75 m on a flat road if the coefficient of static friction between tires and road is 0.80?(m/s)

Is this result independent of the mass of the car?

Answer 1

Concepts and reason

The concepts used to solve this problem are a centripetal force and frictional force.

Equate the centripetal force with frictional force to get the maximum speed. Obtain the expression for the speed. From this relation, the dependency of the speed of the object on the mass can be easily checked. Substitute the given values to get the value of the speed.

Fundamentals

The expression for the centripetal force can be written as follows,

$F = \frac{{m{v^2}}}{r}$ …… (1)

Here, $m$ is the mass of the object, $v$ is the speed and $r$ is the radius.

The expression of the frictional force can be expressed as follows,

$f = \mu mg$ …… (2)

Here, $\mu$ is the coefficient of static friction and $g$ is the acceleration due to gravity.

Equate the centripetal force with the frictional force that is holding the car in place by using equation (1) and (2).

$\frac{{m{v^2}}}{r} = \mu mg$

Rearrange the expression to get the value of speed.

$v = \sqrt {r\mu g}$ …… (3)

Calculate the maximum speed.

$v = \sqrt {r\mu g}$

Substitute $75{\rm{ m}}$ for $r$ , $0.80$ for $\mu$ and $9.8{\rm{ m}} \cdot {{\rm{s}}^{ - 2}}$ .

$\begin{array}{c}\\v = \sqrt {\left( {75{\rm{ m}}} \right)\left( {0.80} \right)\left( {9.8{\rm{ m}} \cdot {{\rm{s}}^{ - 2}}} \right)} \\\\ = 24.25{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}\\\end{array}$

Recall, the expression obtained for speed in step 1.

$v = \sqrt {r\mu g}$

Here, in the expression, there is not any term, which is dependent on the mass of the vehicle.

Ans:

The maximum speed of the car is $24.25{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}$ .

The maximum speed of the car is independent of the mass of the car.