Question

A car turns a corner on a slippery road at a constant speed of \(10 \mathrm{~m} / \mathrm{s}\). If the coefficient of friction is \(0.5\), the minimum radius of the arc at which the car turns is meter. (A) 20 (B) 10 (C) 5 (D) 4

Short answer

The minimum radius the car can turn at without sliding is approximately 20.4 meters, so the closest answer among the options is (A) 20.

Step-by-step solution

1. Identify the forces acting on the car

While the car is turning, there are two main forces acting on it: the gravitational force (weight) acting vertically downward, and the friction force acting horizontally to oppose the car's motion. As the car is moving in a curved path, the centripetal force will be provided by the friction force.

2. Calculate the centripetal force required

We need to calculate the centripetal force keeping the car in its circular path. The formula for centripetal force is: \[ F_c = m * a_c \] where \(F_c\) is the centripetal force, \(m\) is the mass of the car, and \(a_c\) is the centripetal acceleration. The centripetal acceleration can be calculated using the formula: \[ a_c = \frac{v^2}{r} \] where \(v\) is the speed of the car, which is 10 m/s, and \(r\) is the radius of the arc.

3. Calculate the maximum friction force

The maximum friction force available to provide the centripetal force is given by: \[ F_f = \mu * m * g \] where \(F_f\) is the friction force, \(\mu\) is the coefficient of friction (0.5), \(m\) is the mass of the car, and \(g\) is the acceleration due to gravity (approximately \(9.8 \mathrm{m/s^2}\)).

4. Set the centripetal force equal to the maximum friction force

Since the centripetal force must be provided by the friction force, we can set the two forces equal to each other: \[ m * a_c = \mu * m * g \] The mass of the car (\(m\)) can be canceled out from both sides, leaving: \[ a_c = \mu * g \] Plug in the given values for \(\mu\) and \(g\) to find \(a_c\): \[ a_c = 0.5 * 9.8 = 4.9 \mathrm{m/s^2} \]

5. Solve for the radius of the arc

Now that we have the centripetal acceleration (\(a_c\)), we can find the radius of the arc by plugging it back into the centripetal acceleration formula: \[ 4.9 = \frac{10^2}{r} \] \[ r = \frac{10^2}{4.9} \] \[ r \approx 20.4 \mathrm{m} \] The minimum radius the car can turn at without sliding is approximately 20.4 meters, so the closest answer among the options is: (A) 20

Key concepts

Centripetal Acceleration

When an object moves along a circular path, its direction is constantly changing, although its speed might remain constant. This constant change in direction requires a continuous acceleration known as centripetal acceleration. It keeps the object moving in a circle instead of in a straight line. The formula to calculate centripetal acceleration is:\[ a_c = \frac{v^2}{r} \]Here, \(v\) represents the velocity of the object moving in the circle, and \(r\) is the radius of that circle. The greater the velocity, the greater the centripetal acceleration needed to maintain the circle.As seen in the problem, this provides the required acceleration to keep the car from skidding off the slippery curve at a speed of 10 m/s.

Friction Force

Friction is the force that opposes motion between two surfaces that are in contact with each other. In the context of a car making a turn, friction between the tires and the road provides the necessary force to change the car's direction. This force acts perpendicular to the gravitational force and is critical in making the vehicle curve. For the car to turn safely, the friction force must be greater than or equal to the necessary centripetal force that enables the car to circumnavigate the curve without slipping. In our exercise, this friction force is actually what supplies the centripetal force, making it essential for executing turns on a slippery road.

Coefficient of Friction

The coefficient of friction, denoted by \( \mu \), measures how easily one object slides over another. It is a dimensionless number that compares the force of friction between two bodies and the force pressing them together.In this scenario concerning a vehicle turning a corner, the coefficient of friction helps determine the maximum friction force that can act between the tires and the road. The maximum frictional force available can be calculated using the formula:\[ F_f = \mu * m * g \]Where:

• \(F_f\) is the force of friction.
• \(\mu\) is the coefficient of friction, given as 0.5.
• \(m\) is the mass of the car, and
• \(g\) is the acceleration due to gravity, approximately \(9.8 \mathrm{m/s^2}\).

This value is critical for determining how tightly the car can turn without losing grip and slipping out of control.

Circular Motion

Circular motion occurs when an object travels along a curved path or circle. This type of motion is characterized by a constant change in direction, which requires an inward force to ensure the path remains circular.In this exercise, the car experiences circular motion as it navigates the curve. The combination of its velocity and the force of friction ensures the car remains in a circular path. A key relationship in circular motion is:\[ F_c = m * a_c \]In this formula:

• \(F_c\) is the centripetal force needed to keep the radius constant.
• \(m\) is the mass of the object.
• \(a_c\) is the centripetal acceleration.

The link between speed, radius, and centripetal forces is crucial in designing safe road bends and understanding vehicle dynamics, especially on problematic surfaces like slippery roads.