A 1550 -kg car moves on a horizontal curved road. If the radius of the curve is 46 m and the coefficient of friction between the tires and road is 0.1, find the maximum speed the car can have so that it may not skid.
To find the maximum speed the car can have without skidding, we need to consider the maximum centripetal force that can be generated by the friction between the tires and the road.
The centripetal force required to keep the car moving in a curved path can be calculated using the formula:
F = (m * v^2) / r
Where:
F is the centripetal force
m is the mass of the car (1550 kg)
v is the velocity of the car
r is the radius of the curved road (46 m)
The maximum centripetal force is equal to the maximum frictional force that can be generated between the tires and the road, which can be calculated using the formula:
F_friction = μ * N
Where:
F_friction is the frictional force
μ is the coefficient of friction (0.1)
N is the normal force
The normal force can be calculated using the formula:
N = m * g
Where:
g is the acceleration due to gravity (9.8 m/s^2)
Substituting the values into the equations, we get:
F_friction = μ * N
= μ * (m * g)
= 0.1 * (1550 kg * 9.8 m/s^2)
= 1519 N
Now we can equate the maximum centripetal force and the maximum frictional force:
F = F_friction
(m * v^2) / r = 1519 N
Solving for v, we get:
v^2 = (F * r) / m
= (1519 N * 46 m) / 1550 kg
= 45.105 m^2/s^2
Taking the square root of both sides, we find:
v = √(45.105 m^2/s^2)
≈ 6.71 m/s
Therefore, the maximum speed the car can have without skidding is approximately 6.71 m/s.
To find the maximum speed at which the car will not skid, we need to consider the centripetal force and the friction force acting on the car.
The centripetal force required to keep the car moving in a circular path is given by the formula:
Fc = m * v^2 / r
Where:
Fc = Centripetal force
m = Mass of the car
v = Velocity of the car
r = Radius of the curve
The maximum friction force that can act on the car without causing it to skid is given by:
Ff_max = µ * (m * g)
Where:
Ff_max = Maximum friction force
µ = Coefficient of friction
g = Acceleration due to gravity (approximately 9.8 m/s^2)
In order for the car not to skid, the centripetal force and the maximum friction force need to be equal.
So, we can equate the two equations to find the maximum speed:
m * v^2 / r = µ * (m * g)
Canceling out 'm' from both sides:
v^2 / r = µ * g
Rearranging the equation for velocity 'v':
v = √(µ * g * r)
Now we can substitute the known values into the equation and calculate the maximum speed:
v = √(0.1 * 9.8 * 46)
v = √(4.61)
v ≈ 2.15 m/s
Therefore, the maximum speed the car can have without skidding is approximately 2.15 m/s.