A turntable rotates counterclockwise at 78 rpm . A speck of dust on the turntable is at 0.46 rad at t=0. What is the angle of the speck at t = 8.1 s ? Your answer should be between 0 and 2π rad.

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23-07-2019
Mathematics

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A turntable rotates counterclockwise at 78 rpm . A speck of dust on the turntable is at 0.46 rad at t=0. What is the angle of the speck at t = 8.1 s ? Your answer should be between 0 and 2π rad.

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sunedutoit7 sunedutoit7 · Answer 1 · 2019-07-28T12:08:14+02:00

Answer: 3.79 rad

Step-by-step explanation:

Step 1 : Convert RPM to radians/second in order to work with the same units throughout the problem.

1 minute has 60 seconds.

Thus 78 RPM ÷ 60 s = 1.3 Revolutions per second.

1 revolution = 2pi radians

1.3 RPS x 2pi = 8.16814 radians per second

Thus 78 RPM = 8.16814 rad/second

Step 2 : Calculate how far the dust speck travelled in 8.1 seconds.

We just calculated that the speck travels 8.16814 rad/s.

Therefore 8.16814 rad/s × 8.1 s = 66.162 rad/s

The dust speck travelled 66.162 radians in 8.1 s.

Step 3: Calculate the angle of the speck after 8.1 s

The dust peck was at 0.46 rad and travelled 66.162 rad thus 0.46+66.162 = 66.622 rad

Step 4: Convert to a value between 0 and 2pi radians.

Minus 10 revolutions in order to get to a value between 0 and 2pi radians.

66.622 - 10(2pi) = 3.79 rad

Answer: The dust particle is at an angle of 3.79 rad

astha8579 astha8579 · Answer 2 · 2022-02-16T07:49:57+01:00

Angles are measure of degree of rotation from some fixed spot.

The angle of the speck of the dust considered at time t = 8.1 seconds is 0.72 radians.

How to calculate angular speed from rotation frequency?

The key usable fact is that a full rotation covers [tex]2\pi[/tex] radians.

Thus, if something rotates 't' times per minute, then we have:

[tex]2\pi t[/tex] radians of angles rotated in one minute.

Thus, the angular speed of that thing is [tex]2\pi t[/tex] radians per minute.

Since for the given case, as the dust is lying on the turntable which is rotating at 78 rpm(rotations per minute), that dust will also rotate with same speed(presumably)

Thus, we have angular speed of dust as

[tex]78 \times 2\pi = 156 \pi[/tex] radian per minute

or

[tex]156\pi/60 = 2.6 \pi[/tex] radians per second (each second is of length of a minute divided by 60)

For t = 0, the angle was 0.46 radians,and after time interval of 8.1 seconds, the angle will be increased by [tex]8.1 \times 2.6\pi \approx 6.62[/tex] radians.

Thus, total angle of the speck of the dust at t = 8.1 s is

[tex]6.62 + 0.46 = 7.08[/tex] radians.

Since we know that after full rotation ( [tex]2\pi = 6.283[/tex] radians), thing come at same place, so we have

[tex]7.08 \: \rm radians = 7.08 - 2\pi \approx 0.72[/tex] radians

Thus,

The angle of the speck of the dust considered at time t = 8.1 seconds is 0.72 radians.

Learn more about angular speed here:

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