Problem 95 - Precalculus

Question

Find the linear speed of a point that moves with constant speed in a circular motion if the point travels along the circle of are length \(s\) in time \(t\). $$s=2 \mathrm{m}, t=5 \mathrm{sec}$$

Step-by-Step Solution

Answer

Linear speed is 0.4 m/s.

Step 1: Understanding the Relationship

To find the linear speed of a point moving in circular motion, we consider that the linear speed (\(v\)) can be defined by the relation: \[v = \frac{s}{t}\] where s is the arc length (distance) and t is the time taken.

Step 2: Input Values

We are given the arc length (s) and time (t), specifically: \[s = 2 \, \mathrm{m}\] \[t = 5 \, \mathrm{sec}\]

Step 3: Applying the Formula

Substitute the given values of s and t into the formula for linear speed: \[v = \frac{2 \mathrm{m}}{5 \mathrm{sec}} \]

Step 4: Calculating the Linear Speed

Perform the division to find v: \[v = 0.4 \, \mathrm{m/s} \]

Key Concepts

Circular Motion

When an object follows a circular path, it is said to be in circular motion. This type of motion is common in our daily lives, such as the rotation of a Ferris wheel or the spinning of car wheels.
Circular motion can be uniform or non-uniform:

• Uniform Circular Motion: The object travels a circular path at a constant speed.
• Non-uniform Circular Motion: The object's speed changes over time as it travels around the circle.

The motion is dictated by two main components:

• Velocity, which has magnitude (speed) and direction (tangential to the circle).
• Centripetal force, required to keep the object following the circular path.

Understanding circular motion is crucial when calculating linear speed, as the linear speed is closely tied to how fast the point moves along the arc, the curved path the object takes.

Arc Length

Arc length is a vital concept in understanding circular motion. It refers to the distance that a point travels along the curve of a circle. In simpler terms, it's like measuring the segment of the outer edge of a pizza slice.
The arc length is directly connected to the radius and angle of the circle:

• Formula: When an angle \( \theta \) is measured in radians, the arc length \( s \) is given by \( s = r \theta \), where \( r \) is the radius of the circle.

However, in our specific exercise, we're primarily interested in the arc length \( s = 2 \, \mathrm{m} \) given directly to solve for linear speed. This concept is essential as it allows us to translate the rotational movement into a linear perspective for easier understanding and calculation.

Constant Speed

In the context of circular motion, constant speed indicates a scenario where an object maintains the same rate of motion as it travels along its circular path. While the direction of an object may change constantly (as it moves along a circular path), the speed or rate at which it moves does not fluctuate.
Key things to remember about constant speed in circular motion:

• It ensures the velocity's magnitude remains unchanged throughout the journey.
• However, since the direction changes, the velocity is not constant.
• Calculating speed becomes straightforward as speed is simply the arc length divided by time, given by \( v = \frac{s}{t} \).

For the given exercise, knowing the subject is moving at constant speed simplifies finding the linear speed because computations need not account for acceleration or changes in speed. Simply apply the formula \( v = \frac{2 \, \mathrm{m}}{5 \, \mathrm{sec}} = 0.4 \, \mathrm{m/s} \). This approach highlights how straightforward solving for linear speed can be in the case of constant speed in circular motion.