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.

查看解析

登录即可查看完整答案

类似问题

An electrical motor spins at a constant 2695.0 rpm. If the armature radius is 7.165 cm, what is the acceleration of the edge of the rotor?

You are taking a turn at 23.0 m/s on a ramp of radius 39.0 m. What is your acceleration?

An object travels in a circular path of radius r at a constant speed v. What happens to the object’s acceleration if the speed is doubled and the radius stays unchanged?

Two satellites are in orbit around Mars with the same orbital radius and same velocity. Satellite 2 has twice the mass of satellite 1. The radial acceleration of satellite 2 has

An object moves in a circle of radius R at a constant speed v. Is its acceleration zero?

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

People born without creativity never can develop it

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单