A car whose speed is 90.0 km/h (25 m/s) rounds a curve 180 m in radius that is properly banked for speed of 45 km/h (12.5 m/s). Find the minimum coefficient of friction between tires and road that will permit the car to make a turn. What will happen to the car in this case?
My teacher said we'll use 'tan' but i'm confused. :(
The projections of the forces acting ob the car moving at the velocity v=12.5 m/s are:
x: ma=N•sin α,
y: 0=N•cos α – mg.
The normal force has a horizontal component , pointing toward the center of the curve. Because the ramp is to be designed so that the force of static friction is zero, only the component causes the centripetal acceleration:
m•v²/R=N•sin α= m•g•sin α/cos α =m•g•tan α,
tan α= v²/g•R
α =arctan(v²/g•R)= arctan(12.5²/9.8•180) =5.06º
The equations of the motion of the car moving with velocity V=25 m/s are:
x: ma= N•sin α +F(fr) •cos α,
y: 0=N•cos α – mg- F(fr) •sin α.
Since F(fr) =μ•N, we obtain
ma= N•sin α + μ•N •cos α,
mg =N•cos α – μ•N •sin α.
ma/mg =N(sin α + μ •cos α)/N (cos α – μ •sin α)
a=g(sin α + μ •cos α)/ (cos α – μ •sin α)
But
a=V²/R,
therefore,
V²/R =g(sin α + μ •cos α)/ (cos α – μ •sin α),
V²•cos α -R•g•sin α =μ• (V²• sin α +R•g• cos α),
μ=( V²•cos α -R•g•sin α)/ (V²• sin α +R•g• cos α)=
=(625•0.996-180•9.8•0.088/(625•0.088+180•9.8•0.996)=
= 0.258
Thanks, Elena, good work. My analysis was too quick.
To solve this problem, we can use the concept of centripetal force. When a car is moving in a curved path, it experiences a centripetal force that keeps it in that path. For a car traveling at constant speed around a curve, this centripetal force is provided by the friction between the tires and the road.
The maximum frictional force that can be exerted between the tires and the road is given by the equation:
Frictional force (F_friction) = Normal force (F_normal) * Coefficient of friction (μ)
The normal force is the force exerted perpendicular to the surface of contact, which is equal to the weight of the car (mg), where m is the mass of the car and g is the acceleration due to gravity.
Now, let's analyze the forces acting on the car in the given scenario. The car is traveling on a properly banked curve, which means that the banking angle is chosen such that no friction is needed to keep the car on the curve at a specific speed. In this case, the normal force provides all the centripetal force required.
To determine the minimum coefficient of friction required, we need to calculate the centripetal force acting on the car. The centripetal force is given by:
Centripetal force (F_centripetal) = (mass of the car * velocity^2) / radius of the curve
Substituting the given values:
mass of the car = m
velocity = 25 m/s
radius of the curve = 180 m
We have:
F_centripetal = (m * (25 m/s)^2) / 180 m
Since the car is traveling at a speed of 25 m/s, which is greater than the design speed of 12.5 m/s, we can assume that the car is moving faster than intended for the banked curve. In this case, the car will have to rely on both the normal force and the coefficient of friction to make the turn.
Now, to find the minimum coefficient of friction required, we can equate the centripetal force with the sum of the frictional force and the component of the weight acting horizontally towards the center of the curve:
F_centripetal = F_friction + component of weight horizontally towards the center
F_centripetal = μ * F_normal + mg * sin(θ)
Where θ is the angle of banking.
Since we know that the car is moving faster than intended for the banked curve, the frictional force (μ * F_normal) needs to be greater than zero. Therefore, we can set the minimum coefficient of friction to zero and solve for the angle of banking θ.
By rearranging the equation, we get:
θ = arcsin((F_centripetal - μ * F_normal) / mg)
Now, we can calculate the angle of banking θ using the given values and substitute it back into the equation to solve for the minimum coefficient of friction.
Once we find the value of the minimum coefficient of friction, we can compare it with the range of possible values for the coefficient of friction, which typically lies between 0 and 1. If the calculated value is within this range, it means that the car can make the turn. If the calculated value is greater than 1, it means that the car cannot make the turn without exceeding the maximum possible friction.
In summary, to solve this problem, you need to:
1. Calculate the centripetal force using the formula F_centripetal = (m * v^2) / r, where m is the mass of the car, v is the velocity of the car, and r is the radius of the curve.
2. Use the equation θ = arcsin((F_centripetal - μ * F_normal) / mg) to find the angle of banking θ.
3. Substitute the value of θ back into the equation to solve for the minimum coefficient of friction.
4. Compare the calculated value of the minimum coefficient of friction with the range of possible values (typically between 0 and 1) to determine if the car can make the turn.
I hope this explanation helps in understanding the problem and how to solve it.