A 1180 kg car rounds a circular turn of radius 19.1 m. If the road is flat and the coefficient of static friction between the tires and the road is 0.67, how fast can the car go without skidding?
m/s
force friction=forcecentripetal
mg*mu=m v^2/r
v= sqrt (mu*R*g)
To determine how fast the car can go without skidding, we need to calculate the maximum speed at which the centripetal force provided by friction equals the maximum frictional force.
The maximum frictional force can be found using the formula:
\(f_{\text{max}} = \mu \cdot N\),
where \(f_{\text{max}}\) is the maximum frictional force, \(\mu\) is the coefficient of static friction, and \(N\) is the normal force.
In this case, since the road is flat, the normal force is equal to the weight of the car, which is \(m \cdot g\), where \(m\) is the mass of the car and \(g\) is the acceleration due to gravity.
So, \(N = m \cdot g\). Substituting this into the formula for the maximum frictional force, we get:
\(f_{\text{max}} = \mu \cdot m \cdot g\).
The maximum frictional force provides the centripetal force, which can be calculated using the formula:
\(F_{\text{centripetal}} = \frac{{m \cdot v^2}}{{r}}\),
where \(F_{\text{centripetal}}\) is the centripetal force, \(m\) is the mass of the car, \(v\) is the velocity of the car, and \(r\) is the radius of the circular turn.
Setting the maximum frictional force equal to the centripetal force, we can solve for the maximum speed:
\(\mu \cdot m \cdot g = \frac{{m \cdot v^2}}{{r}}\).
Canceling out the mass of the car, we have:
\(\mu \cdot g = \frac{{v^2}}{{r}}\).
We can rearrange this equation to solve for the maximum speed, \(v\):
\(v = \sqrt{{\mu \cdot g \cdot r}}\).
Now we can substitute the given values:
\(\mu = 0.67\),
\(g = 9.8 \, \text{m/s}^2\) (approximate value for acceleration due to gravity),
\(r = 19.1 \, \text{m}\).
Plugging these values into the equation, we get:
\(v = \sqrt{{0.67 \cdot 9.8 \cdot 19.1}}\).
Evaluating this expression, we find:
\(v \approx 10.03 \, \text{m/s}\).
Therefore, the car can go approximately 10.03 m/s without skidding.