A 1500-kg car is making a turn with a 100.0-m radius on a road where the coefficient of static friction is 0.70. What is the maximum speed the car can go without skidding?
To determine the maximum speed the car can go without skidding, we need to find the maximum centripetal force that can be exerted on the car without exceeding the static friction force.
The centripetal force required to keep the car moving in a circle is given by the equation:
Fc = (m * v^2) / r
where Fc 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 turn.
The maximum static friction force (Fs) that can be exerted on the car is given by the equation:
Fs = μ * N
where μ is the coefficient of static friction and N is the normal force. The normal force is equal to the weight of the car, which is given by:
N = m * g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting N into the equation for Fs, we have:
Fs = μ * m * g
Setting Fc equal to Fs, we can solve for the maximum velocity:
Fc = Fs
(m * v^2) / r = μ * m * g
Canceling out m from both sides, we have:
v^2 / r = μ * g
Solving for v, we get:
v = √(μ * g * r)
Plugging in the given values, we have:
μ = 0.70
g = 9.8 m/s^2
r = 100.0 m
Calculating the maximum speed, we get:
v = √(0.70 * 9.8 m/s^2 * 100.0 m)
v = √(686 m^2/s^2 * 100.0 m)
v = √(68600 m^3/s^2)
v ≈ 261.5 m/s
So, the maximum speed the car can go without skidding is approximately 261.5 m/s.
To determine the maximum speed the car can go without skidding, we need to consider the frictional force acting on the car during the turn.
The maximum static frictional force (F_max) can be calculated using the formula:
F_max = (coefficient of static friction) * (normal force)
The normal force (N) can be calculated using the formula:
N = mass * g
where mass is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s²).
Let's plug in the values:
Mass of car (m) = 1500 kg
Radius of turn (r) = 100.0 m
Coefficient of static friction (μ) = 0.70
Acceleration due to gravity (g) = 9.8 m/s²
First, calculate the normal force:
N = m * g
N = 1500 kg * 9.8 m/s²
N = 14700 N
Now, calculate the maximum static frictional force:
F_max = μ * N
F_max = 0.70 * 14700 N
F_max ≈ 10290 N
The maximum static frictional force provides the centripetal force required for the car to make the turn without skidding. The centripetal force (F_c) is given by:
F_c = (mass * velocity²) / radius
Rearranging the equation, we can solve for the maximum velocity (v_max):
v_max = √((F_max * radius) / mass)
Plugging in the values:
v_max = √((10290 N * 100.0 m) / 1500 kg)
v_max ≈ √(6860000 m²/s²) ≈ 2618 m/s
Therefore, the maximum speed the car can go without skidding is approximately 2618 m/s.