Insect on spinning record problem

[lab-rat]

Homework Statement

In an old record player, the flat round vinyl disc (record) is placed on a turntable which spins around. Once it gets going around and around, it moves so that the number of revolutions it makes per minute is constant (33 and 1/3). An insect settles on the edge of the spinning record. It decides to crawl in a straight line to the centre of the record.

a) draw the actual path the insect makes as seen by you the observer
b)If the insect moves from the edge to the centre (a distance of 15 cm) at 1.5cm/s, estimate the arc length of the path the observer sees.

The Attempt at a Solution

For a), the answer is obviously an arc. It would be ) if the disk is spinning clockwise correct?

As for b) I don't really understand. I can't seem to find the formula to be able to find the length of the arc. Since it definitely won't be in the shape of half of a circle, I can't use the formula for the circumference of a circle.. Any suggestions?

 

[PeterO]

Homework Statement

In an old record player, the flat round vinyl disc (record) is placed on a turntable which spins around. Once it gets going around and around, it moves so that the number of revolutions it makes per minute is constant (33 and 1/3). An insect settles on the edge of the spinning record. It decides to crawl in a straight line to the centre of the record.

a) draw the actual path the insect makes as seen by you the observer
b)If the insect moves from the edge to the centre (a distance of 15 cm) at 1.5cm/s, estimate the arc length of the path the observer sees.

The Attempt at a Solution

For a), the answer is obviously an arc. It would be ) if the disk is spinning clockwise correct?

As for b) I don't really understand. I can't seem to find the formula to be able to find the length of the arc. Since it definitely won't be in the shape of half of a circle, I can't use the formula for the circumference of a circle.. Any suggestions?

Firstly it is a sad reflection to see my record player called "an old record player".

If the insect simply sat on the rim of the record for 1 minute, it would travel 33 1n 1/3 revolutions, an arc length 33 and 1/3 times the circumference of the circle.

At the speed given, the insect will take less than a minute to reach the centre, but still the record will have gone through several rotations.

For each rotation, the length of the spiral arc can be taken as the average of a circumference of the circle of largest radius and the circumference of the circle of smallest radius.

That should be enough information to get you going.

 

[lab-rat]

So the circumference of the record is 94.25 cm, which means its arc length of the record is 3141.67 cm?

1 rotation would take 1.8 s so the insect would advance 2.7 cm per rotation right?

I'm sorry, I'm just really lost right now! I've been working on this number for way too long without any examples or explanations from the prof. I haven't done any physics in 2 years and that was at my french high school. All of the new terminology is really confusing so I am very sorry if I'm not catching on very quickly

 

[PeterO]

So the circumference of the record is 94.25 cm, which means its arc length of the record is 3141.67 cm?

1 rotation would take 1.8 s so the insect would advance 2.7 cm per rotation right?

I'm sorry, I'm just really lost right now! I've been working on this number for way too long without any examples or explanations from the prof. I haven't done any physics in 2 years and that was at my french high school. All of the new terminology is really confusing so I am very sorry if I'm not catching on very quickly

Those figures look appropriate.
By the end of 1.8 seconds, the insect is "on a circle" of radius 12.3 cm - which will have a smaller circumference. The spiral track it will have mapped out can be approximated to the average of the outer circumference and the inner circumference. Then there is the next "loop" of the spiral, and the next etc, then finally a fraction of the last "loop"

 

[lab-rat]

Is this what you meant?

 

[PeterO]

Is this what you meant?

This looks good, except it looks like the last part of the spiral is only part of a full rotation.
The ant was 1.5 cm from the centre, so took only 1 second to get there. The record takes 1.85 seconds for a full rotation, 1 second is only 1/1.85 of the full spiral so I would be multiplying that last 4.71 cm by 1/1.85.

Just as an aside - you could have done it in one go.
In all there is 5.54 rotations done during the walk.
Large circle is the record, r = 15
small circle is the centre, r = 0

answer is 5.54 * 2pi * (15+0)/2 which gives the answer you will get once you adjust for that last part "circle".

 

[lab-rat]

I fixed it and ended up with 261.81 which is pretty close to the answer.
How should I formulate my answer? Is Arc length = 261.81 cm appropriate?

Thank you so much for your help by the way, very much appreciated!

 

[PeterO]

I fixed it and ended up with 261.81 which is pretty close to the answer.
How should I formulate my answer? Is Arc length = 261.81 cm appropriate?

Thank you so much for your help by the way, very much appreciated!

The questioner referred to this spiral path as an arc length, so to call it an arc length is appropriate. best to use their term even if it is wrong.

You have to be careful how you round off any values on the way through the problem.
When I don't round off until the final answer I get 261.799 or 261.8 cm