A car rounds a curve that is banked inward. The radius of curvature of the road is R = 109, the banking angle is θ = 25°, and the coefficient of static friction is μs = 0.24. Find the minimum speed that the car can have without slipping.

To find the minimum speed that the car can have without slipping, we need to consider the forces acting on the car as it rounds the curved road. The two main forces at play are the gravitational force (mg) and the frictional force (static friction) between the tires and the road.

First, let's consider the gravitational force. It can be broken down into two components: one perpendicular to the road surface (mgcosθ) and one parallel to the road surface (mgsinθ).

Next, let's consider the frictional force. It acts in the opposite direction to the direction of motion of the car and can also be broken down into two components: one perpendicular to the road surface (N) and one parallel to the road surface (μsN).

Since the car is not slipping, the frictional force has to provide the necessary centripetal force to keep the car moving in a circular path. The centripetal force is given by mv²/R, where m is the mass of the car and v is its speed.

Now, let's set up the equations:

1) For the vertical direction:
N + mgcosθ = 0 (since the car is not moving up or down)

2) For the horizontal direction:
μsN = mgsinθ

3) For the centripetal force:
μsN = mv²/R

We can substitute N from equation 2 into equation 3:

μs(mgcosθ) = mv²/R

Now, we need to solve for v.

Rearranging the equation, we get:

v² = μs(Rgcosθ)

v = sqrt(μs(Rgcosθ))

Plugging in the given values:

v = sqrt(0.24(109)(9.8)cos(25°))

v ≈ 22.4 m/s

Therefore, the minimum speed that the car can have without slipping is approximately 22.4 m/s.

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