Question

A \(2400-\mathrm{lb}(=10.7-\mathrm{kN})\) car traveling at \(30 \mathrm{mi} / \mathrm{h}(=13.4 \mathrm{~m} / \mathrm{s})\) attempts to round an unbanked curve with a radius of \(200 \mathrm{ft}\) \((=61.0 \mathrm{~m}) .(a)\) What force of friction is required to keep the car on its circular path? ( \(b\) ) What minimum coefficient of static friction between the tires and road is required?

Short answer

The force of friction necessary to keep the car on its circular path can be found by first calculating the mass of the car, then substituting this and the given values for speed and radius into the equation \(F = m \cdot v^2 / r\). Then, the minimum coefficient of friction can be found by substituting the force of friction and the weight of the car into the equation \(\mu = F / N\).

Step-by-step solution

Calculate the force required for circular motion

Start by using Newton's second law for circular motion in the horizontal direction: the net force is equal to the mass of the object times its acceleration. In this case, the acceleration is the centripetal acceleration needed to keep the car on its circular path, which is given by \(a = v^2 / r\), where \(v\) is the speed of the car and \(r\) is the radius of the circular path. So, the force needed to keep the car on its circular path is \(F = m \cdot v^2 / r\) where \(m\) is the mass of the car. Substitute the given values into this equation to find the force.

Calculate the mass of the car

The weight of the car is given as 2400 lb, which is a force. In order to use this in the Newton's second law, it needs to be converted to mass. As we know, weight is equal to mass times gravity (\(W = m \cdot g\)). By knowing the acceleration due to gravity (\(g = 9.81 m/s^2\)) and the weight of the car, we can find the mass by \(m = W / g\).

Find the force of friction

Once the mass of the car is found, it can be substituted back into the equation found in step 1. The result will give the force required to keep the car on its circular path, which is the force of friction.

Calculate the minimum coefficient of static friction

The force of friction acting between two surfaces can be found using the equation \(F = \mu \cdot N\), where \(F\) is the force of friction, \(\mu\) is the coefficient of static friction, and \(N\) is the normal force. In this case, the normal force is equal to the weight of the car. Therefore, the coefficient of static friction can be found by solving the equation for \(\mu\), resulting in \(\mu = F / N\). Substitute the force of friction found in Step 3 and the weight of the car for \(N\) into this equation to find the minimum coefficient of static friction.

Key concepts

Centripetal Acceleration

Centripetal acceleration is a key concept when understanding circular motion. Imagine you are in a car rounding a curve; you feel a force pulling you towards the center of the curve. This is due to centripetal acceleration, necessary to change the direction of motion of an object traveling in a circular path. It is directed towards the center of the circle and can be calculated using the formula: \( a = \frac{v^2}{r} \), where \( v \) is the velocity of the object, and \( r \) is the radius of the circular path.

When a car travels around an unbanked curve, centripetal acceleration makes sure the car turns appropriately with the curve. Without sufficient centripetal force, which in most cases is provided by the force of friction between the car's tires and the road, the car would not follow the circular path and could potentially skid outwards.

Force of Friction

The force of friction plays an integral role in circular motion, especially when we consider vehicles navigating turns. Friction is the resistive force that occurs when two surfaces move across each other. It can either aid in motion, such as providing the grip needed for a car to traverse a corner, or impede it, as with a heavy box dragged across the floor.

In the context of a car on a curve, the force of friction acts centripetally, meaning it points towards the center of the circle along which the car is moving. This force is crucial for keeping the car on its path without slipping. Notably, the maximum force of friction available before slipping occurs is dictated by the coefficient of static friction and the normal force, which, for a car on flat ground, is equal to its weight. Higher friction means a greater ability for the car to maintain its circular path at higher speeds.

Coefficient of Static Friction

The coefficient of static friction is a dimensionless number representing the ratio of the force of static friction between two surfaces to the normal force pressing them together. It measures how resistant a stationary object is to starting moving when a force is applied. This value is crucial in scenarios where friction is required to prevent motion, such as a car parked on a hill or, as in our example, keeping a car from slipping while navigating a turn.

On a practical level, knowing the coefficient of static friction allows engineers to understand and predict whether a particular material combination (like car tires on road surfaces) will be able to withstand certain forces without slipping. If a car's tires have a high enough coefficient of static friction with the road surface, the car can make turns at higher speeds without losing grip.

Newton's Second Law

Newton's second law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration (\( F = m \times a \) ). This law is pivotal in explaining how forces cause objects to move and is fundamental when analyzing the car's motion on a circular path. It tells us that for a car to maintain circular motion, a centripetal force must act upon it; this force results from the aforementioned force of friction.

When we apply Newton's second law to circular motion, it bridges the concepts of force, mass, and centripetal acceleration, informing us how much frictional force is needed to keep a vehicle like the car in the example on its path. It also plays a critical role in determining the maximum safe speed at which a vehicle can navigate a turn given the coefficient of static friction between the tires and the road.