A typical laboratory centrifuge rotates at 4000 rpm. Test tubes have to be placed into a centrifuge very carefully because of the very large accelerations. For comparison, what is the magnitude of the acceleration a test tube would experience if dropped from a height of 1.0 m and stopped in a 1.0-ms-long encounter with a hard floor?

검증된 단계별 안내

Step 1: Start by calculating the velocity of the test tube just before it hits the floor. Use the kinematic equation for free fall:

v = \sqrt{2 g h}

, where

g

is the acceleration due to gravity (9.8 m/s²) and

h

is the height (1.0 m).

Step 2: Once the velocity

v

is determined, calculate the deceleration (negative acceleration) experienced by the test tube when it comes to a stop. Use the formula:

a = \frac{v}{t}

, where

t

is the stopping time (1.0 ms or 0.001 s).

Step 3: Substitute the value of

v

from Step 1 and the stopping time

t

into the formula for

a

to calculate the magnitude of the deceleration.

Step 4: Recognize that the deceleration is much larger than the acceleration due to gravity, which is why the test tube must be handled carefully to avoid damage.

Step 5: Compare the calculated deceleration to the acceleration experienced in the centrifuge (from part a of the problem, if available) to understand the relative magnitudes of these forces.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Centrifugal Acceleration

Centrifugal acceleration is the apparent force that draws a rotating object away from the center of rotation, experienced by objects in a rotating system, such as a centrifuge. It is calculated using the formula a = ω²r, where ω is the angular velocity in radians per second and r is the radius of the circular path. In a centrifuge, this acceleration can be significantly larger than gravitational acceleration, leading to the need for careful placement of test tubes.

Free Fall and Impact Acceleration

When an object is dropped from a height, it accelerates due to gravity until it impacts the ground. The acceleration during free fall is approximately 9.81 m/s², but the impact acceleration can be much greater depending on how quickly the object comes to a stop. In this scenario, the test tube experiences a rapid deceleration over a short time (1.0 ms), which can result in a very high magnitude of acceleration upon impact.

Impulse and Momentum

Impulse is the change in momentum of an object when a force is applied over a period of time. It is calculated as the product of the average force and the time duration of the force application. In the context of the test tube dropping and stopping, the impulse experienced during the brief encounter with the floor is crucial for determining the force exerted on the test tube and the resulting acceleration, which can be calculated using the impulse-momentum theorem.