Problem 20 - An Introduction to Physical Science

Question

A person drives a car around a circular, level cloverleaf with a radius of \(70 \mathrm{m}\) at a uniform speed of \(10 \mathrm{m} / \mathrm{s}\). (a) What is the acceleration of the car? (b) Compare this answer with the acceleration due to gravity as a percentage. Would you be able to sense the car's acceleration if you were riding in it?

Step-by-Step Solution

Answer

(a) Acceleration is 1.43 m/s². (b) It's 14.59% of gravity; noticeable when riding.

Step 1: Understand the Problem

We are given a car moving in a circular path with radius \( r = 70 \, \text{m} \) at a speed of \( v = 10 \, \text{m/s} \). We need to find the acceleration of the car and compare it to the acceleration due to gravity.

Step 2: Calculate Centripetal Acceleration

The formula for centripetal acceleration \( a_c \) is \( a_c = \frac{v^2}{r} \). Substitute \( v = 10 \, \text{m/s} \) and \( r = 70 \, \text{m} \) into the formula: \[ a_c = \frac{10^2}{70} = \frac{100}{70} \approx 1.43 \, \text{m/s}^2 \].

Step 3: Compare with Gravitational Acceleration

The acceleration due to gravity is \( g = 9.8 \, \text{m/s}^2 \). To find the centripetal acceleration as a percentage of gravity, calculate \( \left(\frac{a_c}{g}\right) \times 100 \): \[ \frac{1.43}{9.8} \times 100 \approx 14.59\% \].

Step 4: Determine Sensibility of Acceleration

Humans can typically feel accelerations of a few meters per second squared. Since the car's acceleration is approximately 14.59% of gravity, this would be noticeable to someone in the car, as it is over 1 m/s².

Key Concepts

Centripetal Force

When moving along a circular path at a constant speed, an object experiences centripetal force. This is the force directed towards the center of the circle, responsible for keeping the object on its circular path.

Centripetal force is not a new type of force, but rather the net result of other forces acting on an object to cause circular motion. For example, in the case of a car driving around a curve, it could be friction between the car's tires and the road that provides the centripetal force.

To find the required centripetal force, use the equation:

• \[F_c = m \cdot a_c\]

Here, \(F_c\) is the centripetal force, \(m\) is the mass of the object, and \(a_c\) is the centripetal acceleration. Understanding this concept is essential for analyzing any object moving in a circle.

Acceleration due to Gravity

Acceleration due to gravity is a constant force that pulls objects towards the Earth's center. This force is experienced by all objects when they are near the Earth's surface and is denoted by \(g\).

The standard value for the acceleration due to gravity on Earth is approximately \(9.8 \, \text{m/s}^2\).

This value can change slightly depending on altitude and geographical location, but for most practical purposes, \(9.8 \, \text{m/s}^2\) is used.

• \(g\) affects objects in free fall, where it causes them to accelerate downwards.
• It also affects objects in circular motion, as it provides a reference point for calculating how different accelerations compare to gravity.

When comparing an acceleration to gravity, it helps understand how noticeable it would be, given that \(g\) is a constant, familiar force we experience daily.

Circular Motion

Circular motion refers to the movement of an object along the perimeter of a circle. This can either be at constant speed (uniform circular motion) or varying speeds (non-uniform circular motion).

In uniform circular motion, although the speed is constant, the velocity is not. Since velocity is a vector quantity, and any change in direction results in a change in velocity, the object is always accelerating. This specific acceleration is called centripetal acceleration, and it is always directed towards the center of the circle.

Key aspects of circular motion include:

• The radius of the circular path: Determines the curvature of the path an object is taking.
• Speed of the object: Although constant, plays a crucial role in calculating centripetal acceleration and force.
• Centripetal Acceleration: Given by \(a_c = \frac{v^2}{r}\), it depends on the speed \(v\) and the radius \(r\) of the path.

Mastering these components is vital for solving problems related to objects moving in circles.