Respuesta :
The vinyl record, which rotates at 45 rpm, has an angular velocity of 4.71 radians per second, and has a period 1.33 seconds.
Further explanation
Movement is the change of position of an object or a point, movement can be slipt into 2 basic motions, translation and rotation. In the case of our problem, we have pure rotation, since the vinyl record doesn't move anywhere, it sits still while the disc rotates over an axis.
In the problem, we are asked to compute the angular velocity of the vinyl record. We know that the vinyl record makes 45 revolutions per minute, this already is a measure of the angular velocity, however it is a custom to express it in radians per second. Therefor to answer the question we need to change its dimensions, we know that 1 revolution is equal to [tex]2 \pi[/tex] radians, and 1 minute is equal to 60 seconds, therefor:
[tex]45 \frac{rev}{min} = 45 \cdot \frac{2 \pi}{60} \frac{rad}{s} = 4.71 \frac{rad}{s}[/tex]
Now let's compute the period. The period is a measure of how much time does a rotating object complete a revolution, and it's computed through the following formula:
[tex]T= \frac{2 \pi}{w}[/tex]
Where [tex]w[/tex] is the angular velocity. By plugging values we obtain that the period is equal to 1.33 seconds.
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Keywords
Angular velocity, rotation, period of motion
(a)The angular velocity of the single-play vinyl record is [tex]\boxed{\frac{{3\pi }}{2}\,{{{\text{rad}}} \mathord{\left/{\vphantom {{{\text{rad}}} {\text{s}}}} \right.\kern-\nulldelimiterspace} {\text{s}}}}[/tex] or [tex]\boxed{4.71\,{{{\text{rad}}} \mathord{\left/{\vphantom {{{\text{rad}}} {\text{s}}}} \right.\kern-\nulldelimiterspace} {\text{s}}}}[/tex].
(b)The time period of rotation of the vinyl record is [tex]\boxed{\frac{4}{3}\,{\text{s}}}[/tex] or [tex]\boxed{1.333\,{\text{s}}}[/tex].
Further Explanation:
Part (a):
The rpm is the measurement of the number of rotations made by the body in one minute. The angular velocity of a body in the unit of radian per second is the measure of the angle covered by the body in one second.
The angular velocity of the body in radian per second is given by:
[tex]\omega = \left( {rpm} \right) \times \left( {\frac{{2\pi }}{{60}}} \right){{rad} \mathord{\left/ {\vphantom {{rad} {\sec }}} \right.\kern-\nulldelimiterspace} {\sec }}[/tex]
Substitute the value of rpm in the above expression.
[tex]\begin{aligned}\omega&= \left( {45\,{{{\text{rev}}} \mathord{\left/ {\vphantom {{{\text{rev}}} {{\text{min}}}}} \right.\kern-\nulldelimiterspace} {{\text{min}}}}} \right) \times \left( {\dfrac{{2\pi }}{{60}}} \right){{{\text{rad}}} \mathord{\left/{\vphantom {{{\text{rad}}} {\text{s}}}} \right.\kern-\nulldelimiterspace} {\text{s}}}\\&=\dfrac{3\pi}{2}\,\text{rad/s}\\&\approx4.71\,\text{rad/s}\end{aligned}[/tex]
Thus, the angular speed of rotation of the vinyl record in radian per second is [tex]\boxed{\frac{{3\pi }}{2}\,{{{\text{rad}}} \mathord{\left/{\vphantom {{{\text{rad}}} {\text{s}}}} \right.\kern-\nulldelimiterspace} {\text{s}}}}[/tex] or [tex]\boxed{4.71\,{{{\text{rad}}} \mathord{\left/{\vphantom {{{\text{rad}}} {\text{s}}}} \right.\kern-\nulldelimiterspace} {\text{s}}}}[/tex].
Part (b):
The time period of rotation of a body is the time taken by the body in completing one revolution. The time period of rotation of the vinyl record is given by:
[tex]T = \dfrac{{2\pi }}{\omega }[/tex] …… (I)
Substitute [tex]\dfrac{{3\pi }}{2}\,{{{\text{rad}}} \mathord{\left/{\vphantom {{{\text{rad}}} {\text{s}}}} \right.\kern-\nulldelimiterspace} {\text{s}}}[/tex] for [tex]\omega[/tex] in above expression.
[tex]\begin{aligned}T&= \dfrac{{2\pi }}{{\left( {\dfrac{{3\pi }}{2}} \right)}}\,{\text{s}} \\&={\text{}}\dfrac{4}{3}\,{\text{s}} \\&={\text{ 1}}{\text{.333}}\,{\text{s}} \\\end{aligned}[/tex]
Thus, the time period of the vinyl record is [tex]\boxed{\frac{4}{3}\,{\text{s}}}[/tex] or [tex]\boxed{1.333\,{\text{s}}}[/tex].
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Answer Details:
Grade: College
Chapter: Circular motion
Subject: Physics
Keywords: Old fashioned, single play vinyl record, rotates, angular velocity, time period of rotation, revolution per minute, rpm, radian per second.
