A typical laboratory centrifuge rotates at 7,598 rotations per minute. Test tubes have to be placed into the centrifuge very carefully because of the large acceleration. What is the magnitude of the acceleration (in units of
×
10
4
m/s
2
) at the end of the test tube that is 0.073 m from the axis of rotation? Enter the numerical value only, do not enter units. Round to the nearest second decimal place.

Question

A typical laboratory centrifuge rotates at 7,598 rotations per minute. Test tubes have to be placed into the centrifuge very carefully because of the large acceleration. What is the magnitude of the acceleration (in units of 
×
10
4
 m/s
2
) at the end of the test tube that is 0.073 m from the axis of rotation?  Enter the numerical value only, do not enter units. Round to the nearest second decimal place.

BlackTom AI · Accepted Answer

Question restatement: A centrifuge spins at 7,598 revolutions per minute, with test tubes located 0.073 m from the axis. You are asked to find the magnitude of the centripetal (radial) acceleration at the end of the test tube, expressed in units of 10^4 m/s^2, and to enter only the numerical value (no units), rounded to two decimal places.

Step 1: Convert the rotational speed to angular velocity. The angular velocity ω is related to the rotation rate in revolutions per minute by ω = 2π × (rpm/60).
- Compute rpm/60: 7598 / 60 ≈ 126.6333 rev/s.
- Multiply by 2π: ω ≈ 2π × 126.6333 ≈ 795.7 rad/s.

Step 2: Use the centripetal acceleration formula a = ω^2 r. Here r = 0.073 m.
- First compute ω^2: (795.7 rad/s)^2 ≈ 633,000 (approximately 6.33 × 10^5) s^−2.
- Multiply by r: a ≈ 633,000 × 0.073 ≈ 46,200 m/s^2.

Step 3: Express the acceleration in the requested units. The problem asks for units of 10^4 m/s^2, so divide by 10,000:
- a/(10^4) ≈ 46,200 / 10,000 ≈ 4.62.
- Round to two decimal places: 4.62.

Step 4: Check for consistency. Recomputing with slightly more precise ω (using 7598/60 = 126.6333..., and ω = 126.6333 × 2π) yields ω ≈ 795.66 rad/s, ω^2 ≈ 633,083 s^−2, and a ≈ 633,083 × 0.073 ≈ 46,216 m/s^2, which gives a/(10^4) ≈ 4.62 as well. Thus the result 4.62 is consistent with the calculation.

Notes on potential confusion:
- If one mistakenly rounds ω too early or uses an approximate rpm-to-rad/s conversion, a could appear near 46,000–46,400 m/s^2, which would yield 4.60–4.64 in the requested units. Careful use of the exact conversion leads to 4.62.
- The problem explicitly asks for the numerical value without units and in units of 10^4 m/s^2, so the final entry should be 4.62.

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