Question

Calculate the centripetal force exerted on a vehicle of mass \(m=1500 .\) kg that is moving at a speed of \(15.0 \mathrm{~m} / \mathrm{s}\) around a curve of radius \(R=400 . \mathrm{m} .\) Which force plays the role of the centripetal force in this case?

Short answer

Question: Calculate the centripetal force exerted on a vehicle moving in a curve with a speed of 15.0 m/s and radius of 400 m, and identify which force plays the role of the centripetal force. The mass of the vehicle is 1500 kg. Answer: To calculate the centripetal force, first find the centripetal acceleration using the formula \(a_c = \frac{v^2}{R}\), where \(v = 15.0\) m/s and \(R = 400\) m. Next, use the formula \(F_c = m \times a_c\) to calculate the centripetal force. In this case, the friction force between the tires and the road surface plays the role of the centripetal force.

Step-by-step solution

Calculate the centripetal acceleration

We already have the speed of the vehicle (\(v = 15.0\) m/s) and the radius of the curve (\(R = 400\) m). Using the centripetal acceleration formula, calculate the centripetal acceleration: $$ a_c = \frac{v^2}{R} $$

Calculate the centripetal force

Now that we have the centripetal acceleration, we can calculate the centripetal force using the formula \(F_c = m \times a_c\). We know the mass of the vehicle is \(m=1500\) kg. So, $$ F_c = m \times a_c $$

Identify which force plays the role of the centripetal force

In the case of a vehicle moving in a curve, the friction force between the tires and the road surface is the force that plays the role of the centripetal force. The friction force provides the necessary force to keep the vehicle moving in a circular path and maintain the vehicle's speed while turning.

Key concepts

Centripetal Acceleration

When an object travels in a circle, it constantly accelerates towards the center of that circle, even if it maintains a constant speed. This phenomenon is known as centripetal acceleration. Imagine swirling a ball attached to a string; the ball moves in a circular path because of the inward force you provide through the string.

Mathematically, centripetal acceleration (\(a_c\)) can be calculated using the formula \[a_c = \frac{v^2}{R}\] where \(v\) represents the object's velocity and \(R\) is the radius of the circular path. In the exercise, the vehicle's velocity is given as \(15.0\) m/s, and it's moving in a circular path with a radius of \(400\) m. Substituting these values into the formula gives us the vehicle's centripetal acceleration while turning.

Circular Motion

Circular motion is the movement of an object along the circumference of a circle or rotation along a circular path. It's characterised by a change in direction at every point of motion, which means there is a continuous change in velocity, and consequently, a continuous acceleration known as centripetal acceleration.

The object's constant speed reflects a steady tangential velocity, while its changing velocity indicates a steady acceleration towards the circle's center. Importantly, for an object to maintain circular motion, there must be a net inward force acting on it, creating the centripetal acceleration. This net inward force is the centripetal force. For the vehicle in the exercise, circular motion is achieved as it travels around the curve, constantly changing its direction.

Friction Force

Friction is the resistive force that occurs when two surfaces slide or attempt to slide across each other. It is crucial in everyday life and is the force that allows us to walk without slipping and cars to move without skidding.

In the context of circular motion, the friction force between the vehicle's tires and the road surface plays a critical role. It acts as the centripetal force that allows the vehicle to follow the circular path without slipping outwards due to inertia. The frictional force is dependent on both the nature of the tires and the road surface as well as the normal force, which in this case is the weight of the vehicle. Without sufficient friction, the vehicle would not be able to make the turn and would continue moving in a straight line.