Question

an ultracentrifuge accelerates from rest to 100,000 rpm in 2.00 min. (a) what is the average angular acceleration in ? (b) what is the tangential acceleration of a point 9.50 cm from the axis of rotation? (c) what is the centripetal acceleration in and multiples of g of this point at full rpm? (d) what is the total distance traveled during the acceleration by a point 9.5 cm from

Answer 1

The angular acceleration of an ultracentrifuge can be calculated by converting the rpm to rad/s and then using the formula for angular acceleration. Tangential and centripetal accelerations are then determined based on the ultracentrifuge's radius and angular velocity, with appropriate unit conversions.

Step-by-step explanation:

Calculation of Ultracentrifuge Accelerations

We're asked to solve several problems related to a rapidly accelerating ultracentrifuge in terms of angular and tangential accelerations, as well as centripetal acceleration. Angular acceleration is the rate of change of angular velocity, and it can be calculated using the formula α = Δω / Δt, where α represents angular acceleration, Δω is the change in angular velocity, and Δt is the period over which the change occurs. Tangential acceleration is related to angular acceleration by the formula at = α * r, where r is the radius from the axis of rotation. Centripetal acceleration, on the other hand, refers to the acceleration directed towards the centre of the circular path, calculated using ac = ω^2 * r, where ω is the angular velocity.

To solve for the angular acceleration of the ultracentrifuge that accelerates from rest to 100,000 rpm in 2.00 minutes, we need to first convert rpm to radian per second (rad/s) since the SI unit of angular acceleration is rad/s². Once the conversion is done, we use the angular acceleration formula to find the average α. The tangential acceleration can then be found by multiplying α by the given radius, 9.50 cm (converted to meters). The centripetal acceleration at full rpm is found using the formula given above, with the final angular velocity. This can also be expressed in multiples of the acceleration due to gravity (g ~ 9.81 m/s²).

Answer 2

The average angular acceleration of the ultracentrifuge can be calculated by finding the difference between the final and initial angular velocities divided by the time. The tangential acceleration of a point 9.50 cm from the axis of rotation can be found using the formula radius times angular acceleration. The centripetal acceleration at full rpm can be found using the formula radius times angular velocity squared. The total distance traveled during the acceleration can be found using the formula for distance with uniformly accelerated motion.

Step-by-step explanation:

(a) To find the average angular acceleration, we can use the formula:

Angular acceleration = (final angular velocity - initial angular velocity) / time

Given that the initial angular velocity is 0 rpm, the final angular velocity is 100,000 rpm, and the time is 2.00 min (or 120 s), we can calculate:

Angular acceleration = (100,000 rpm - 0 rpm) / 120 s

(b) The tangential acceleration of a point 9.50 cm from the axis of rotation can be found using the formula:

Tangential acceleration = radius * angular acceleration

Given that the radius is 9.50 cm (or 0.095 m), and the angular acceleration is the value calculated in part (a), we can substitute these values to find the tangential acceleration.

(c) The centripetal acceleration of a point at full rpm can be found using the formula:

Centripetal acceleration = radius * angular velocity^2

Given that the radius is 9.50 cm (or 0.095 m), and the angular velocity is the final angular velocity calculated in part (a), we can substitute these values to find the centripetal acceleration.

(d) To find the total distance traveled during the acceleration by a point 9.50 cm from the axis of rotation, we can use the formula:

Total distance = initial velocity * time + 0.5 * acceleration * time^2

Given that the initial velocity is 0 (since the point starts from rest), the time is 2.00 min (or 120 s), and the acceleration is the tangential acceleration calculated in part (b), we can substitute these values to find the total distance.