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Chapter 4: Problem 7

A car traveling at 30 mph encounters a curve in the road. The radius of the road curve is \(100 \mathrm{ft}\). Find the maximum speeds (mph) before losing traction, if the coefficient of friction on a dry road is \(\mu_{\mathrm{dry}}=0.7\) and on a wet road is \(\mu_{\text {wet }}=0.3\)

Short Answer

Expert verified
The maximum speeds before losing traction on dry and wet roads are approximately 44.64 mph and 25.98 mph, respectively.

Step by step solution

01

Understanding centripetal force

Centripetal force is the force that keeps a body following a curved path. Its magnitude is given by the formula \(F_c = m v^2 / r\), where \(m\) is the mass of the car, \(v\) is the speed of the car, and \(r\) is the radius of the curve.
02

Understanding frictional force

The maximum static friction force between the car tires and the road helps the car make the turn. This force is given by \(F_f = \mu m g\), where \(\mu\) is the coefficient of static friction, \(m\) is the mass of the car, and \(g\) is acceleration due to gravity.
03

Setting up the equation

A car remains in circular motion as long as the maximum static friction force is greater than or equal to the centripetal force. So, we set these two forces equal to establish the maximum speed before losing traction: \(F_f = F_c \Rightarrow \mu m g = m v^2 / r\).
04

Solving for maximum speed on dry road

Rearranging the equation for \(v\), we have \(v = \sqrt{\mu g r}\). Substituting the given values for dry road \(\mu_{dry} = 0.7\), \(r = 100 ft = 30.48 m\) (as 1 ft = 0.3048 m), and \(g = 9.8 m/s^2\), we find the maximum speed.
05

Solving for maximum speed on wet road

Similar to step 4, for the wet road, we use \(\mu_{wet} = 0.3\) and solve for the maximum speed.
06

Conversion to mph

The maximum speeds obtained in the previous steps will be in \(m/s\). Convert them to mph by multiplying by 2.237 (Since 1 m/s = 2.237 mph) to get the final solutions.

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