Question

A bicycle is moving with a speed of \(4.02 \mathrm{~m} / \mathrm{s}\). If the radius of the front wheel is \(0.450 \mathrm{~m}\), how long does it take for that wheel to make a complete revolution? a) \(0.703 \mathrm{~s}\) b) \(1.23 \mathrm{~s}\) c) \(2.34 \mathrm{~s}\) d) \(4.04 \mathrm{~s}\) e) \(6.78 \mathrm{~s}\)

Short answer

Answer: a) The time taken for one complete revolution of the front wheel is approximately 0.703 seconds.

Step-by-step solution

Find the bicycle's speed

Given the bicycle's speed is \(4.02 \mathrm{~m} / \mathrm{s}\). This means that the outer edge of the front wheel also moves at this speed.

Find the radius of the front wheel

The radius of the front wheel is given as \(0.450 \mathrm{~m}\).

Calculate the circumference of the front wheel

To find the circumference (distance covered in one complete revolution), we need to use the formula \(C=2\pi r\) where \(C\) is the circumference, \(\pi\) is a constant approximately equal to \(3.1416\), and \(r\) is the radius. In this case, the radius is \(0.450 \mathrm{~m}\). \(C=2\pi r = 2 \times 3.1416 \times 0.450 \mathrm{~m} \approx 2.83 \mathrm{~m}\).

Determine the time taken for one complete revolution

Now that we have the circumference and the speed, we can find the time taken for the wheel to make one complete revolution using the formula: \(time = \frac{distance}{speed}\), where \(distance\) is the circumference of the wheel and \(speed\) is the bicycle's speed. Substituting the values, we get: \(time = \frac{2.83 \mathrm{~m}}{4.02 \mathrm{~m} / \mathrm{s}} \approx 0.703 \mathrm{~s}\). So the correct answer is a) \(0.703 \mathrm{~s}\).

Key concepts

Circumference of a Circle

The circumference of a circle is one of the fundamental concepts in geometry and can be often found in physics problems related to circular motion. It refers to the distance around the edge of the circle. To calculate it, we use the formula:

\[C = 2\text{\textpi}r\],

where \(C\) represents the circumference, \(\text{\textpi}\) is Pi - a mathematical constant approximately equal to 3.1416, and \(r\) is the radius of the circle, which is the distance from the center to any point on its edge.
In our bicycle wheel example, with the radius of 0.450 meters, we apply the formula to find out how far the wheel travels in one full rotation. It's important to understand that this circumference is also the distance that the bicycle travels forward with every full spin of the wheel. By learning to connect the concept of circumference with physical movement, students can better grasp how geometry is used to describe motion in our day to day life.

Angular Velocity

Angular velocity is a measure of how fast an object rotates or revolves relative to another point, which we say is the angular speed of the object. In the context of uniform circular motion, angular velocity is typically expressed in radians per second (rad/s). It is given by the formula:

\[ \text{\textomega} = \frac{\theta}{t} \],

where \(\text{\textomega}\) represents the angular velocity, \(\theta\) is the angle in radians through which a point or line has been rotated in a specific time duration \(t\). Remember that one complete revolution is \(2\text{\textpi}\) radians.
For example, if our bicycle wheel makes one complete revolution, it rotates through an angle of \(2\text{\textpi}\) radians. If we know the time taken to complete this revolution from our problem, we can calculate the angular velocity. This term is crucial for understanding rotational dynamics in physics and is also applied widely in engineering concepts such as motor speeds, and wheel rotations.

Uniform Circular Motion

Uniform circular motion describes an object moving in a circular path with constant speed. The velocity of an object in uniform circular motion is constantly changing due to its direction, but the speed (the magnitude of velocity) is constant. This concept is vital in understanding various physical phenomena and in designing mechanical systems that undergo circular paths.

Objects in uniform circular motion experience an inward force called centripetal force that keeps them moving in the circle. This force is always perpendicular to the displacement of the object and thus does not do work—meaning it does not change the kinetic energy of the object, and consequently, the speed remains constant. Understanding this helps students see why the speed – the scalar quantity – is constant, but the velocity – a vector quantity which includes direction – is always changing due to the object's continuous change in direction.
The next time you see an object like the wheel of a bicycle traveling in its path, try to visualize the principles of uniform circular motion in action. Comprehending the principle of uniform circular motion lays the foundation not just for solving physics problems but also in appreciating the forces and motion in the physical world around us.