Problem 36 - College Physics

Question

A car with a constant speed of \(83.0 \mathrm{~km} / \mathrm{h}\) enters a circular flat curve with a radius of curvature of \(0.400 \mathrm{~km} .\) If the friction between the road and the car's tires can supply a centripetal acceleration of \(1.25 \mathrm{~m} / \mathrm{s}^{2},\) does the car negotiate the curve safely? Justify your answer.

Step-by-Step Solution

Answer

No, the car cannot safely negotiate the curve; the required centripetal acceleration exceeds the available frictional force.

Step 1: Understanding the Problem

The car moves at a constant speed of 83.0 km/h and enters a circular path with a radius of 0.400 km. We need to determine if the frictional force is sufficient to provide the required centripetal acceleration to keep the car on its path, which is given as 1.25 m/s².

Step 2: Convert Speed to Meters per Second

Convert the car's speed from km/h to m/s using the conversion factor (1 km/h = 0.27778 m/s): \[ 83.0 \text{ km/h} \times 0.27778 = 23.05 \text{ m/s} \]

Step 3: Calculate Required Centripetal Acceleration

Use the formula for centripetal acceleration: \[ a_c = \frac{v^2}{r} \] where \( v = 23.05 \text{ m/s} \) and \( r = 400 \text{ m} \).\[ a_c = \frac{(23.05)^2}{400} \approx 1.33 \text{ m/s}^2 \]

Step 4: Compare with Maximum Available Acceleration

Compare the calculated centripetal acceleration (1.33 m/s²) with the given maximum frictional acceleration (1.25 m/s²). Since 1.33 m/s² > 1.25 m/s², the available friction is not sufficient to provide the required centripetal force.

Step 5: Conclusion

Since the maximum frictional acceleration of 1.25 m/s² is less than the required centripetal acceleration of 1.33 m/s², the car cannot safely negotiate the curve at the given speed.

Key Concepts

Understanding Circular Motion

Circular motion occurs when an object follows a circular path. In this type of motion, the object constantly changes direction. This means there's acceleration involved even if the object's speed remains constant.
This acceleration is known as centripetal acceleration, which is directed towards the center of the circle. It's essential because it keeps the object moving in its curved path.

• Circular motion is common in many real-world situations, such as cars negotiating curves, planets orbiting stars, and more.
• The necessary force that maintains circular motion — the centripetal force — must come from sources like gravity, tension, or friction, depending on the context.

Role of Frictional Force

Frictional force plays a critical role in enabling vehicles to maintain circular motion on a curved path. It's the force that prevents cars from slipping off the road when they're making turns.
Friction arises from the contact between the car's tires and the road surface. This force acts towards the center of the circular path, providing the centripetal force needed.

• On a flat road, the friction between the tires and the road surface is the primary source of centripetal force.
• If the required centripetal force exceeds the available frictional force, the vehicle may skid or slide off the curve.
• The coefficient of friction and the normal force affect the amount of frictional force available.

Connecting Kinematics to Motion

Kinematics involves describing the motion of objects using concepts like velocity, displacement, and acceleration. In the context of circular motion, velocity is always changing direction.
For an object moving at a constant speed in a circle, the velocity is tangent to the circle, while the acceleration points toward the center. This central acceleration ensures that the object continues its circular path.

• The kinematic equations can help analyze the motion's characteristics, such as time taken to complete a circle or changes in velocity.
• In the case given, the concepts of kinematics are applied to understand the change in direction due to circular motion.

Exploring Acceleration

Acceleration signifies any change in the velocity of an object. It can be a change in the speed, direction, or both. In circular motion, even if speed remains constant, the changing direction implies acceleration.
This special type of acceleration, called centripetal acceleration, is vital for maintaining circular paths. It is calculated using the formula:\[ a_c = \frac{v^2}{r} \]where:

• \( v \) is the velocity of the object, and
• \( r \) is the radius of the circular path.

• Higher speeds or smaller curves require greater centripetal acceleration.
• In our example, understanding this concept highlights why the available frictional force wasn't enough to safely negotiate the curve.