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Chapter 11: Problem 20

Vinyl Record A vinyl record on a turntable rotates at \(33 \frac{1}{3}\) rev/min. (a) What is its rotational speed in radians per second? What is the translational speed of a point on the record at the needle when the needle is (b) \(15 \mathrm{~cm}\) and (c) \(7.4 \mathrm{~cm}\) from the turntable axis?

Short Answer

Expert verified
Rotational Speed: 3.49 radians/sec. Translational Speed at 15 cm: 0.5235 m/s. Translational Speed at 7.4 cm: 0.25826 m/s.

Step by step solution

01

Calculate Rotational Speed in Radians per Second

First, convert the rotational speed from revolutions per minute to radians per second. Given the speed is \(33 \frac{1}{3} \) revolutions per minute, it can be written as \(33.33 \) rev/min. Use the conversion factor \(1 \text{ rev} = 2\pi \text{ radians} \) and \(1 \text{min} = 60 \text{ sec} \): \[ \text{Rotational Speed} = 33.33 \left( \frac{2 \pi}{60} \right) = 3.49 \text{ radians/sec} \]So, the rotational speed is approximately \(3.49 \text{ radians/sec} \).
02

Calculate Translational Speed when Needle is 15 cm from Axis

The translational speed can be found using the formula \[ v = r \omega \] where \(r = 15 \text{ cm} = 0.15 \text{ m} \) and \( \omega = 3.49 \text{ radians/sec} \). \[ v = 0.15 \times 3.49 = 0.5235 \text{ m/s} \]So, the translational speed is approximately \( 0.5235 \text{ m/s} \).
03

Calculate Translational Speed when Needle is 7.4 cm from Axis

Similarly, use the same formula \[ v = r \omega \] where \(r = 7.4 \text{ cm} = 0.074 \text{ m} \) and \( \omega = 3.49 \text{ radians/sec} \). \[ v = 0.074 \times 3.49 = 0.25826 \text{ m/s} \]So, the translational speed is approximately \( 0.25826 \text{ m/s} \).

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Key Concepts

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

Rotational Speed
Rotational speed refers to how fast an object is spinning around a central axis. It is usually measured either in revolutions per minute (rev/min) or radians per second (rad/s). A vinyl record on a turntable spins at a certain rotational speed which we need to convert from rev/min to rad/s for practical use. If a record rotates at 33 1/3 rev/min, we write it as 33.33 rev/min. We then use the conversion factor: 1 revolution equals \(2\pi\) radians and 1 minute equals 60 seconds. This gives us a rotational speed of approximately 3.49 radians per second. This speed helps us understand how swiftly the record is spinning.
Translational Speed
Translational speed is the linear speed of a point located at a certain distance from the axis of rotation. To find the translational speed of a point on a spinning vinyl record (such as where the needle sits), we use the formula \[ v = r \, \omega \] where \(v\) is the translational speed, \(r\) is the radius (distance from the turntable axis), and \(\omega\) is the rotational speed in radians per second. For example, if the needle is 15 cm from the axis and the rotational speed is 3.49 rad/s, we find \( v = 0.15 \, m \times 3.49 \, rad/s \approx 0.5235 \, m/s \). The translational speed helps visualize how fast a specific point on the record is moving in a straight line.
Radians Per Second
When discussing rotational motion, radians per second (rad/s) is a crucial unit of measurement. One revolution covers \(2\pi\) radians because the circumference of a circle in radians is \(2\pi\). So, converting revolutions per minute to radians per second involves multiplying by \(2\pi\) and then dividing by 60. This unit tells us how many radians a point on the rotating object covers in one second. For example, for a vinyl record rotating at 33.33 rev/min, converting it to rad/s results in \(33.33 \times \frac{2\,\pi}{60}\approx 3.49\). This rate helps comparing rotational speed across different objects and contexts.
Conversion of Units
Often in physics, we need to convert between different units to make calculations easier or to maintain consistency. Here, converting the rotational speed of a vinyl record from revolutions per minute to radians per second involves understanding that 1 rev = \(2\pi\) radians and 1 min = 60 sec. Hence, when we have 33 1/3 rev/min, writing it as 33.33 rev/min and applying the formula: \( 33.33 \times \frac{2\pi}{60} \), we get approximately 3.49 rad/s. This kind of unit conversion helps us tackle a variety of problems in rotational and translational motion, ensuring accuracy and consistency in our results.

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Most popular questions from this chapter

Polar Axis of Earth (a) What is the rotational speed \(\omega\) about the polar axis of a point on Earth's surface at a latitude of \(40^{\circ} \mathrm{N} ?\) (Earth rotates about that axis.) (b) What is the translational speed \(v\) of the point? What are (c) \(\omega\) and \((\mathrm{d}) v\) for a point at the equator?

A Disk A disk, initially rotating at \(120 \mathrm{rad} / \mathrm{s}\), is slowed down with a constant rotational acceleration of magnitude \(4.0 \mathrm{rad} / \mathrm{s}^{2} .(\mathrm{a})\) How much time does the disk take to stop? (b) Through what angle does the disk rotate during that time?

Hands of a Clock What is the rotational speed of (a) the second hand, (b) the minute hand, and (c) the hour hand of a smoothly running analog watch? Answer in radians per second.

Cylinder Rotates about Horizontal A uniform cylinder of radius \(10 \mathrm{~cm}\) and mass \(20 \mathrm{~kg}\) is mounted so as to rotate freely about a horizontal axis that is parallel to and \(5.0 \mathrm{~cm}\) from the central longitudinal axis of the cylinder. (a) What is the rotational inertia of the cylinder about the axis of rotation? (b) If the cylinder is released from rest with its central longitudinal axis at the same height as the axis about which the cylinder rotates, what is the rotational speed of the cylinder as it passes through its lowest position?

Turntable Two A record turntable is rotating at \(33 \frac{1}{3}\) rev/min. A watermelon seed is on the turntable \(6.0 \mathrm{~cm}\) from the axis of rotation. (a) Calculate the translational acceleration of the seed, assuming that it does not slip. (b) What is the minimum value of the coefficient of static friction, \(\mu^{\text {stat }}\), between the seed and the turntable if the seed is not to slip? (c) Suppose that the turntable achieves its rotational speed by starting from rest and undergoing a constant rotational acceleration for \(0.25 \mathrm{~s}\). Calculate the minimum \(\mu^{\text {stat }}\) required for the seed not to slip during the acceleration period.
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