Physics 4a Collected Homework 2 Please Do Not Attempt To Wri

Physics 4a Collected Homework 2please Do Not Attempt To Write Your So

Some road engineers in Montgomery, Alabama built a banked turn for a circular cloverleaf style highway entrance ramp. Unfortunately, they misread the blueprints and built the turn so that the slope banked outward instead of inward. If the road had been built properly, the turn would have made an angle of 25° with the horizontal, tilted towards the center of the turn. Instead, the road tips away from the center at 25° to the horizontal. The radius of the turn is 50 m. The coefficient of static friction between tires and the road is μs = 0.8. Draw a force diagram for the car on the turn, including the static friction force, normal force, and weight. Label the direction of the net force needed for the car to maintain constant speed and radius. Calculate the maximum speed at which a car can go around this turn without slipping outward.

Paper For Above instruction

The problem involves analyzing the forces acting on a vehicle navigating a banked turn with an abnormal outward tilt, and determining the maximum safe velocity to prevent slipping due to insufficient friction. This requires understanding the dynamics of circular motion, the components of forces on inclined planes, and the role of static friction in maintaining the vehicle’s trajectory.

To begin, the force diagram must include the weight of the car (mg), the normal force (N) exerted by the road surface, and the static friction force (f_s). Since the road is inclined outward at 25° relative to the horizontal, the normal force acts perpendicular to the inclined surface, directed outward. The static friction force acts parallel to the surface, opposing slipping toward the outside of the turn. The weight acts vertically downward. When the car moves at constant speed, the net force component toward the center of the circle provides the necessary centripetal acceleration.

In a coordinate system aligned with the inclined surface, the weight (mg) can be split into components parallel and perpendicular to the surface. The normal force balances the perpendicular component, while the static friction provides the additional centripetal force needed to keep the car on the curved path. For maximum speed without slipping, the static friction force reaches its maximum value, f_s = μs N.

Resolving forces, the maximum allowable static friction force is μs N, where N = mg / cos(25°), since the normal force is related to the weight component perpendicular to the surface. The maximum centripetal force is then f_c = m v^2 / r.

The force balance equations lead to the maximum velocity:

f_s + N sin(25°) = (m v^2) / r

Considering the maximum static friction (f_s = μs N), solving for v yields:

v_max = sqrt[ r g (μs cos(25°) + sin(25°)) ]

Plugging in the values: r = 50 m, g = 9.81 m/s², μs = 0.8, and θ = 25°:

v_max = sqrt[50 9.81 (0.8 * cos(25°) + sin(25°))]

Calculating the trigonometric functions:

cos(25°) ≈ 0.9063

sin(25°) ≈ 0.4226

Evaluating the expression inside the square root:

50 9.81 (0.8 0.9063 + 0.4226) ≈ 50 9.81 (0.725 + 0.4226) ≈ 50 9.81 1.1476 ≈ 50 11.243

v_max ≈ sqrt(562.15) ≈ 23.7 m/s

Therefore, the maximum speed is approximately 23.7 m/s. If the car exceeds this velocity, the static friction limit is surpassed, and the tires will slip outward, risking loss of control.

References

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