Question

A point deseribes a cirele with constant velocity. Show that the velocity with which its projection moves along a given diameter is proportional to the distance of the point from this diameter.

Short answer

Question: Show that the velocity of a point's projection on a given diameter of a circle is proportional to the distance of the point from that diameter, considering that the point is moving with constant angular velocity. Answer: The velocity of the point's projection along the given diameter is proportional to the distance of the point from the diameter because when we divide the linear velocity of the projection by the distance from the diameter, we get a constant expression. Specifically, we found that $\frac{\frac{d(OP')}{dt}}{d} = \frac{\cos(\theta) * \omega}{\sin(\theta)}$, which consists of constant factors.

Step-by-step solution

Set up the problem

First, let's assume a circle with radius R and a point P on the circle, where P is moving with constant angular velocity ω. Let's draw a diameter of the circle and project the point P onto this diameter, calling this point P'. We will also draw a line from the center (O) of the circle to P. The angle from the given diameter to the line OP is θ.

Find the relation between P and P'

The distance from the point P to the given diameter is equal to the perpendicular distance of P and P'. So, from right triangle OPP', we have: $$ d = R * \sin(\theta) $$ where d is the distance from P to the given diameter.

Calculate the velocity of the point P

Since the point is moving with constant angular velocity ω, we can calculate the linear velocity v of P by using the formula: $$ v = R * \omega $$

Calculate the velocity of the projection P'

The projection of the point P onto the diameter is moving along the diameter. In order to calculate the velocity of P', we can use the right triangle OPP'. From the properties of the right triangle, we have: $$ \sin(\theta) = \frac{OP'}{OP} $$ Since \(OP = R\), we have: $$ OP' = R * \sin(\theta) $$ We can use this relation to find the linear velocity of P' along the diameter. Differentiating both sides with respect to time t, we get: $$ \frac{d(OP')}{dt} = R * \frac{d(\sin(\theta))}{dt} $$ The left side is the linear velocity of P' along the diameter, and we need to find the derivative of \(\sin(\theta)\) with respect to time. Using the chain rule, we have: $$ \frac{d(\sin(\theta))}{dt} = \cos(\theta) * \frac{d(\theta)}{dt} $$ Since the angular velocity is constant, \(\frac{d(\theta)}{dt} = \omega\), and we get: $$ \frac{d(OP')}{dt} = R * \cos(\theta) * \omega $$

Show that the velocity of P' is proportional to the distance from P to the given diameter

From step 2, we have $$ d = R * \sin(\theta) $$ and from step 4, we have $$ \frac{d(OP')}{dt} = R * \cos(\theta) * \omega $$ Dividing both equations, we get: $$ \frac{\frac{d(OP')}{dt}}{d} = \frac{R * \cos(\theta) * \omega}{R * \sin(\theta)} = \frac{\cos(\theta) * \omega}{\sin(\theta)} $$ Since all the factors on the right side of the equation are constants, the velocity of the projection P' along the given diameter is proportional to the distance of the point P from the given diameter.

Key concepts

Constant Angular Velocity

Understanding constant angular velocity is vital when studying circular motion. Imagine an object moving along the edge of a circle—just like the second hand of a regular clock. Now, if the object takes the same amount of time to cover each section of the circle, it's moving with a constant angular velocity. This is commonly denoted by the Greek letter ω (omega). Angular velocity measures how fast the angle (in radians) changes as the object moves along the circular path. If ω stays the same no matter where the object is on the circle, we say its angular velocity is constant.

In such a case, no matter where you look at the moving object, it's always turning through angles at the same rate. This rate doesn't depend on the size of the circle since angular velocity is concerned with the angles swept per unit time, not the actual path length.

Linear Velocity

Moving on to the concept of linear velocity, this measures how fast an object is traveling along a path. In circular motion cases like our textbook problem, a point moving around the circle has a linear velocity that is related to the constant angular velocity. A useful analogy is a car moving around a race track; its speed along the track is its linear velocity.

For circular motion at a constant angular velocity, the linear velocity v can be calculated by the product of the radius R and the angular velocity ω, given by the equation v = R * ω. In layman's terms, the further you are from the center of a merry-go-round, the faster you need to move to keep pace with those closer to the center, even though everyone is rotating at the same rate (constant angular velocity).

Right Triangle Properties

Right triangle properties are the bread and butter of trigonometry and are indispensable in problems involving circular motion. In a right triangle, the Pythagorean theorem helps relate the lengths of the sides, whereas the trigonometric functions relate these lengths to the angles within the triangle. In our exercise, the relationship between the distance of a point from a diameter in the circle and the point's projection on the diameter leverages these properties.

Specifically, the sine function relates the angle θ and the side opposite to the angle (d) to the hypotenuse of the right triangle (R), following the equation d = R * sin(θ). These properties can simplify complex problems by breaking them down into right triangles, where the direct relationships between angles and sides can be easily determined.

Proportional Relationships

Lastly, the idea of proportional relationships is a cornerstone in understanding the relationship between different elements of motion in a uniform circular path. When two quantities are proportional, they vary in such a way that one quantity is a constant multiple of the other, or their ratio is constant. This is exactly what happens in our exercise; the velocity of the projection of the point on the diameter is directly proportional to the distance from the point to the diameter.

If one of these changes by a certain factor, the other changes by the same factor, too. This gives us a very reliable way to predict how one variable will change if we know how the other variable changes. Proportional relationships in this context imply that objects closer to the diameter move slower than those farther away since their velocities decrease as their distances to the diameter decrease.