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Chapter 1: Problem 6

A compact disc (CD) has a diameter of 12 centimeters (cm). Suppose that the CD is in a CD-player and is rotating at 225 revolutions per minute. What is the angular velocity of the CD (in radians per second) and what is the linear velocity of a point on the edge of the CD?

Short Answer

Expert verified
The angular velocity of the CD is approximately 23.56 radians per second, and the linear velocity of a point on the edge of the CD is approximately 141.36 centimeters per second.

Step by step solution

01

Finding the Angular Velocity

To find the angular velocity in radians per second, we will use the following formula: \[Angular\ Velocity = \frac{2 * \pi * Revolutions}{Time}\] Given the number of revolutions per minute, we'll first convert it into revolutions per second: \[225\ revolutions/minute * \frac{1\ minute}{60\ seconds} = 3.75\ revolutions/second\] Now, we'll apply the formula: \[Angular\ Velocity = \frac{2 * \pi * 3.75}{1}\]
02

Calculating the Angular Velocity

Multiply the values to get the angular velocity in radians per second: \[Angular\ Velocity = 2 * 3.14 * 3.75\] \[Angular\ Velocity \approx 23.56\ radians/second\] So, the angular velocity of the CD is approximately 23.56 radians per second.
03

Finding the Linear Velocity

To find the linear velocity (v) of a point on the edge of the CD, we need to use the following formula: \[v = r * Angular\ Velocity\] The diameter of the CD is given as 12 cm, so the radius (r) will be half of that: \[r = \frac{12}{2} = 6\ cm\] Now, substitute the radius and the angular velocity into the formula: \[v = 6 * 23.56\]
04

Calculating the Linear Velocity

Multiply the values to get the linear velocity in centimeters per second: \[v = 6 * 23.56\] \[v \approx 141.36\ cm/second\] So, the linear velocity of a point on the edge of the CD is approximately 141.36 centimeters per second.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Velocity
Linear velocity refers to how fast a point is moving along a straight path. Imagine a point on the edge of a CD; as the CD spins, this point traces a circular path. The linear velocity is the speed of this point along the circular path. This can be calculated using the formula:
  • \[ v = r \times \text{Angular Velocity} \]
Where \( r \) is the radius of the circle and the Angular Velocity is given in radians per second.
For example, if the diameter of a CD is 12 cm, the radius \( r \) is half of that, which is 6 cm. To find the linear velocity when the angular velocity is known, simply plug in the values to the formula.
This calculation helps us understand how fast a point on the edge of a spinning object is moving in terms of distance per unit of time.
In our case, the linear velocity of a point on the CD's edge turned out to be approximately 141.36 cm/second.
This means that if the point were to move along a straight path at this speed, it would cover 141.36 centimeters in one second.
Radians Per Second
Radians per second is a measure of angular velocity, which tells us how quickly something is rotating. A radian is a unit that measures angles based on the radius of a circle. When we say something is moving at a certain number of radians per second, we are indicating how many radian-lengths the object covers in one second.
To find the angular velocity in radians per second from revolutions per minute, follow these steps:
  • Convert revolutions per minute to revolutions per second by dividing by 60.
  • Multiply this by \( 2\pi \) (since there are \( 2\pi \) radians in one full revolution).
In the case of the CD which spins at 225 revolutions per minute, converting this gives us 3.75 revolutions per second. When translated to radians per second, using the formula\[ \text{Angular Velocity} = 2 \times \pi \times 3.75 \],
this gives us approximately 23.56 radians/second. This indicates how swiftly the CD is spinning.
Revolutions Per Minute
Revolutions per minute (RPM) is a way to express the number of turns a rotating object makes around an axis in one minute. It鈥檚 a common unit used to represent the speed of rotation.
In practice, you might see RPMs when dealing with CDs, fans, engines, and other mechanical devices.
To relate RPM to other measurements, you might need to convert it. For example:
  • To obtain revolutions per second, divide the RPM by 60 (since there are 60 seconds in a minute).
  • To convert to radians per second, further multiply by \( 2\pi \).
In our CD example, 225 RPM is converted to 3.75 revolutions per second.
Analyzing RPMs helps in understanding how fast a rotating device or component is functioning relative to its maximum speed or other moving parts.
Understanding RPMs allows one to more easily predict and calculate related motions, like angular or linear speeds.

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Most popular questions from this chapter

A small pulley with a radius of 3 inches is connected by a belt to a larger pulley with a radius of 7.5 inches (See Figure 1.16 ). The smaller pulley is connected to a motor that causes it to rotate counterclockwise at a rate of 120 rpm (revolutions per minute). Because the two pulleys are connected by the belt, the larger pulley also rotates in the counterclockwise direction. (a) Determine the angular velocity of the smaller pulley in radians per minute. (b) Determine the linear velocity of the rim of the smaller pulley in inches per minute. (c) What is the linear velocity of the rim of the larger pulley? Explain. (d) Find the angular velocity of the larger pulley in radians per minute. (e) How many revolutions per minute is the larger pulley turning?

This exercise provides an alternate method for determining the exact values of \(\cos \left(\frac{\pi}{6}\right)\) and \(\sin \left(\frac{\pi}{6}\right)\). The diagram to the right shows the terminal point \(P(x, y)\) for an arc of length \(t=\frac{\pi}{6}\) on the unit circle. The points \(A(1,0)\), \(B(0,1),\) and \(C(x,-y)\) are also shown. Notice that \(B\) is the terminal point of the \(\operatorname{arc} t=\frac{\pi}{2},\) and \(C\) is the terminal point of the arc \(t=-\frac{\pi}{6}\). We now notice that the length of the arc from \(P\) to \(B\) is $$ \frac{\pi}{2}-\frac{\pi}{6}=\frac{\pi}{3} $$ In addition, the length of the arc from \(C\) to \(P\) is $$ \frac{\pi}{6}-\frac{-\pi}{6}=\frac{\pi}{3} $$ This means that the distance from \(P\) to \(B\) is equal to the distance from \(C\) to \(P\) (a) Use the distance formula to write a formula (in terms of \(x\) and \(y\) ) for the distance from \(P\) to \(B\). (b) Use the distance formula to write a formula (in terms of \(x\) and \(y\) ) for the distance from \(C\) to \(P\). (c) Set the distances from (a) and (b) equal to each other and solve the resulting equation for \(y\). To do this, begin by squaring both sides of the equation. In order to solve for \(y\), it may be necessary to use the fact that \(x^{2}+y^{2}=1\) (d) Use the value for \(y\) in (c) and the fact that \(x^{2}+y^{2}=1\) to determine the value for \(x\). Explain why this proves that $$\cos \left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2} \text { and } \sin \left(\frac{\pi}{3}\right)=\frac{1}{2}$$.

In each of the following, when it is possible, determine the exact measure of central the angle in degrees. Otherwise, round to the nearest hundredth of a degree. (a) The central angle that intercepts an arc of length \(3 \pi\) feet on a circle of radius 5 feet. (b) The central angle that intercepts an arc of length 18 feet on a circle of radius 5 feet. (c) The central angle that intercepts an arc of length 20 meters on a circle of radius 12 meters. (d) The central angle that intercepts an arc of length 5 inches on a circle of radius 5 inches. (e) The central angle that intercepts an arc of length 12 inches on a circle of radius 5 inches.

(a) Use a calculator (in radian mode) to determine five-digit approximations for \(\cos (4)\) and \(\sin (4)\) (b) Use a calculator (in radian mode) to determine five-digit approximations for \(\cos (4-\pi)\) and \(\sin (4-\pi)\) (c) Use the concept of reference arcs to explain the results in parts (a) and (b).

A small pulley with a radius of 10 centimeters inches is connected by a belt to a larger pulley with a radius of 24 centimeters inches (See Figure 1.16 ). The larger pulley is connected to a motor that causes it to rotate counterclockwise at a rate of \(75 \mathrm{rpm}\) (revolutions per minute). Because the two pulleys are connected by the belt, the smaller pulley also rotates in the counterclockwise direction. (a) Determine the angular velocity of the larger pulley in radians per minute. (b) Determine the linear velocity of the rim of the large pulley in inches per minute. (c) What is the linear velocity of the rim of the smaller pulley? Explain. (d) Find the angular velocity of the smaller pulley in radians per second. (e) How many revolutions per minute is the smaller pulley turning?
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