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.

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?

A 62.9 kg lady is standing on a weighing scale in an elevator as shown in the diagram below. Assume gravitational acceleration = 15 ms-2 Given that the elevator is accelerating upwards at 3.7 ms-2 , determine the magnitude of the normal force on the lady.

An object is fired from ground level at an angle of 31.9 degrees to the horizontal. Taking acceleration due to gravity = 7.65 ms-2 If the initial speed is 5.6 ms-1, how far from the starting point will it hit the ground? Assume air resistance is negligible.

更多留学生实用工具

智能学习助手

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