Compare the maximum speeds at which a car can safely navigate a flat, unbanked turn of radius 50.0 m in dry conditions versus snowy conditions. The coefficient of static friction in dry weather is 0.90, and in snowy weather it is 0.10.
So, find the maximum speed for each weather condition. The frictional force must equal the centripetal force at the maximum speed. That is,
To find the maximum speed for each weather condition, we need to equate the frictional force to the centripetal force. The frictional force is given by the product of the coefficient of static friction and the normal force, while the centripetal force is given by the mass of the car multiplied by the centripetal acceleration.
In this case, the normal force is equal to the weight of the car, which can be calculated as the mass of the car multiplied by the acceleration due to gravity (9.8 m/s^2). The centripetal acceleration is the square of the velocity divided by the radius of the turn.
For the dry conditions:
Frictional force = Centripetal force
μs * m * g = m * v^2 / r
Where:
μs is the coefficient of static friction (0.90)
m is the mass of the car (not given)
g is the acceleration due to gravity (9.8 m/s^2)
v is the maximum speed
r is the radius of the turn (50.0 m)
Simplifying the equation, we have:
0.90 * m * 9.8 = m * v^2 / 50.0
Rearranging the equation to solve for v, the maximum speed, we get:
v^2 = 50.0 * 0.90 * 9.8
v = √(50.0 * 0.90 * 9.8)
Now, let's calculate the maximum speed for the snowy conditions using the same steps but with a coefficient of static friction of 0.10:
Frictional force = Centripetal force
0.10 * m * g = m * v^2 / 50.0
Rearranging the equation to solve for v, we get:
v^2 = 50.0 * 0.10 * 9.8
v = √(50.0 * 0.10 * 9.8)
Now we can calculate the maximum speeds for each weather condition by evaluating the expressions:
For dry conditions:
v = √(50.0 * 0.90 * 9.8)
For snowy conditions:
v = √(50.0 * 0.10 * 9.8)
Remember to substitute the appropriate values for the variables (mass, coefficient of static friction, and acceleration due to gravity) in order to obtain the precise maximum speeds for each weather condition.