Problem 224 - Calculus Volume 3

Question

A Ferris wheel car is moving at a constant speed \(v\) and has a constant radius \(r\). Find the tangential and normal acceleration of the Ferris wheel car.

Step-by-Step Solution

Answer

Tangential acceleration is 0; normal acceleration is \(\frac{v^2}{r}\).

Step 1: Understanding Tangential Acceleration

The tangential acceleration is the rate of change of the tangential velocity. Since the Ferris wheel is moving at a constant speed, this implies there is no change in speed. Therefore, the tangential acceleration is given by:\[ a_t = \frac{d v}{d t} = 0 \]

Step 2: Understanding Normal (Centripetal) Acceleration

The normal or centripetal acceleration occurs when an object moves in a circular path, directed towards the center of the circle. The formula for normal acceleration is:\[ a_n = \frac{v^2}{r} \]where \(v\) is the linear speed of the car and \(r\) is the radius of the Ferris wheel.

Key Concepts

Tangential Acceleration

Tangential acceleration refers to the acceleration that occurs along the edge of a circular path. It is essentially the change in speed of an object as it moves along the path. In the context of our Ferris wheel example, the car is moving at a constant speed, which means its speed isn’t increasing or decreasing.
Since tangential acceleration is concerned with the change in speed, if the speed is constant, the tangential acceleration is zero. Mathematically, this is expressed as:

• \( a_t = \frac{d v}{d t} = 0 \)

Here \( a_t \) stands for tangential acceleration, and \( \frac{d v}{d t} \) represents the rate of change in speed over time. Since there’s no change in speed for our Ferris wheel car, the tangential acceleration remains zero.

Normal Acceleration

Normal acceleration, also known as centripetal acceleration, appears whenever an object moves along a circular path. It acts perpendicular (normal) to the direction of motion, pulling the object toward the center of the circular path. This is critical for maintaining circular motion.
In the case of the Ferris wheel car, which moves in a circular path, the normal acceleration comes into play. It ensures the car stays on its curved path without veering off. The formula for normal acceleration is shown below:

• \( a_n = \frac{v^2}{r} \)

Here:

• \( a_n \) is the normal (centripetal) acceleration.
• \( v \) is the constant speed of the Ferris wheel car.
• \( r \) is the radius of the Ferris wheel.

This formula helps determine how much force is needed to keep the Ferris wheel car traveling in a circle.

Constant Speed

Constant speed is a term that signifies that an object is moving at the same rate over time, without speeding up or slowing down. In the Ferris wheel scenario, this means the speed of the car is unchanged as it moves around the circle.
The importance of constant speed in circular motion lies in its impact on tangential acceleration; specifically, that there is none. Constant speed

• ensures no increase or decrease in the magnitude of velocity in the tangential direction,
• leads to zero tangential acceleration.

Always remember: when speed remains constant, tangential acceleration disappears, but the circular motion continues thanks to the unchanging normal (centripetal) acceleration helping to maintain the path.

Centripetal Acceleration

Centripetal acceleration is vital to understanding how objects move in a circular path. Unlike tangential acceleration, which affects speed changes, centripetal acceleration does not change the speed but changes the direction of the object’s velocity toward the circle's center.
For the Ferris wheel car, the centripetal acceleration, also called normal acceleration, can be calculated using:

• \( a_c = \frac{v^2}{r} \)

where

• \( a_c \) is centripetal acceleration, pointing towards the center of the Ferris wheel.
• \( v \) is the constant velocity of the car.
• \( r \) is the Ferris wheel radius.

This constant inward acceleration is what enables the car to maintain its path along the wheel, ensuring it doesn’t fly off due to tangential momentum. Centripetal acceleration keeps all moving things like planets in orbits and vehicles on curve roads seamlessly in their circular paths.