A car weighing 15,000N rounds a curve of 60m at an angle of 30 degrees. Find the friction force acting on the tires when the car is travelling at 100kph. The coefficient of friction between tires and the road is 0.90
First, convert the speed of the car from kph to m/s:
100 kph * (1000m/1 km) * (1 hr/3600 s) = 27.78 m/s
Next, we will find the critical speed at which the car would slip using the formula:
v_c = sqrt(mu * R * g)
Where v_c is the critical speed, mu is the coefficient of friction, R is the radius of the curve, and g is the acceleration due to gravity (approx. 9.8 m/s^2).
v_c = sqrt(0.90 * 60m * 9.8 m/s^2)
v_c = sqrt(529.2 m^2/s^2)
v_c = 23.0 m/s
Since the car's speed (27.78 m/s) is greater than the critical speed (23.0 m/s), we need to find the required friction force to keep the car from slipping.
First, let's find the horizontal component of the car's weight which can contribute to the required friction force, W_horizontal:
W_horizontal = 15000N * sin(30 degrees)
W_horizontal = 15000N * 0.5
W_horizontal = 7500N
Now, let's find the required friction force:
Specifically, we will use the formula for centripetal force:
F_centr = m * v^2 / R
Where F_centr is the required centripetal force, m is the car's mass (which can be found by dividing the weight by acceleration due to gravity; m = 15000N/9.8m/s^2 = 1531kg), v is the car's speed, and R is the radius of the curve.
F_centr = 1531kg * (27.78m/s)^2 / 60m
F_centr = 1531kg * 772.1m^2/s^2 / 60m
F_centr = 19949 N
Next, we will find the friction force:
F_friction = F_centr - W_horizontal
F_friction = 19949 N - 7500 N
F_friction = 12449 N
When the car is traveling at 100 kph around the curve of 60 meters, the friction force acting on the tires is approximately 12,449 N.
To find the friction force acting on the tires, we need to calculate the normal force and then use the coefficient of friction. Here are the steps to solve the problem:
Step 1: Convert the speed from kilometers per hour (kph) to meters per second (m/s).
The formula to convert from kph to m/s is:
speed in m/s = speed in kph * (1000 / 3600)
speed in m/s = 100 * (1000 / 3600) = 27.78 m/s (rounded to 2 decimal places)
Step 2: Calculate the acceleration of the car.
The formula for acceleration is:
acceleration = v^2 / r
Here, v is the speed of the car and r is the radius of the curve.
acceleration = (27.78^2) / 60
acceleration ≈ 12.87 m/s^2 (rounded to 2 decimal places)
Step 3: Calculate the net force acting on the car.
The formula for net force is:
net force = mass * acceleration
Since weight is equal to mass multiplied by the acceleration due to gravity, we can rewrite this formula as:
net force = weight * acceleration due to gravity * acceleration / weight
Since the weight acts vertically downwards, the vertical component of the weight is balanced by the normal force, and only the horizontal component of the weight contributes to the net force.
net force = weight * sin(angle) * acceleration due to gravity / weight
simplifying, we get:
net force = sin(angle) * acceleration due to gravity
Step 4: Calculate the friction force.
The formula for friction force is:
friction force = coefficient of friction * normal force
To find the normal force, we can use the vertical component of the weight and the equation:
normal force = weight * cos(angle)
Now we can substitute the values and calculate the friction force:
normal force = weight * cos(angle)
normal force = 15000N * cos(30)
normal force ≈ 12990N (rounded to 2 decimal places)
friction force = coefficient of friction * normal force
friction force = 0.90 * 12990N
friction force ≈ 11691N (rounded to 2 decimal places)
Therefore, the friction force acting on the tires is approximately 11691N.
To find the friction force acting on the tires, we need to first calculate the net force acting on the car while it is rounding the curve.
The net force is the difference between the centripetal force (Fc) required to keep the car moving in a circular path and the force due to gravity (Fg) acting on the car:
Net Force = Fc - Fg
The centripetal force (Fc) can be calculated using the formula:
Fc = (m * v²) / r
where m is the mass of the car, v is the velocity of the car, and r is the radius of the curve.
To find the mass of the car, we can use the equation:
Fg = m * g
where Fg is the force due to gravity and g is the acceleration due to gravity (approximately 9.8 m/s²).
Given that the weight of the car is 15,000N, we can calculate the mass (m) using the formula:
m = Fg / g
Now, let's calculate the mass of the car:
m = 15,000N / 9.8 m/s² = 1,530.61 kg
Next, we need to convert the velocity of the car from kph to m/s. To do this, we divide the speed by 3.6:
Velocity (m/s) = 100 kph / 3.6 = 27.78 m/s
Now we can calculate the radius of the curve in meters. Given that the car travels 60m around the curve, we have:
r = 60m
Using these values, we can calculate the centripetal force (Fc):
Fc = (m * v²) / r
Fc = (1,530.61 kg * (27.78 m/s)²) / 60m
Fc ≈ 1,500 N
Now, we can find the net force:
Net Force = Fc - Fg
Net Force = 1,500 N - 15,000 N
Net Force = -13,500 N
The negative sign indicates that the force is acting in the opposite direction of the car's motion.
Finally, to find the friction force (Ff) acting on the tires, we can use the formula:
Ff = μ * Fn
where μ is the coefficient of friction and Fn is the normal force.
The normal force (Fn) can be calculated as:
Fn = m * g
Fn = 1,530.61 kg * 9.8 m/s²
Fn ≈ 15,000 N
Now we can calculate the friction force:
Ff = 0.90 * 15,000 N
Ff = 13,500 N
Therefore, the friction force acting on the tires when the car is traveling at 100 kph is approximately 13,500 N.