(II) A viscometer consists of two concentric cylinders, 10.20 cm and 10.60 cm in diameter. A liquid fills the space between them to a depth of 15.0 cm. The outer cylinder is fixed, and a torque of 0.024 m · N keeps the inner cylinder turning at a steady rotational speed of 57 rev/min. What is the viscosity of the liquid?

Verified step by step guidance

1. Convert the diameters of the cylinders to radii in meters. The radius of the inner cylinder, $r_1$, is half of 10.20 cm, and the radius of the outer cylinder, $r_2$, is half of 10.60 cm.
2. Convert the rotational speed from revolutions per minute (rev/min) to radians per second (rad/s). Use the conversion factor $1 \text{ rev} = 2\pi \text{ rad}$ and $1 \text{ min} = 60 \text{ s}$.
3. Calculate the angular velocity, $\omega$, of the inner cylinder using the formula $\omega = 2\pi \times \text{(revolutions per second)}$.
4. Use the formula for the viscosity $\eta$ of the liquid in a cylindrical viscometer, given by $\eta = \frac{T}{2\pi h (r_2^2 - r_1^2) \omega}$, where $T$ is the torque, $h$ is the depth of the liquid, and $\omega$ is the angular velocity.
5. Substitute the values of $T$, $h$, $r_1$, $r_2$, and $\omega$ into the formula to calculate the viscosity, $\eta$, of the liquid.

Key Concepts

Viscosity

Viscosity is a measure of a fluid's resistance to flow or deformation. It quantifies how thick or sticky a liquid is, affecting how easily it can move. In the context of the viscometer, viscosity is determined by the torque required to maintain a certain rotational speed of the inner cylinder while the liquid is between the cylinders.

Torque

Torque is a measure of the rotational force applied to an object. It is calculated as the product of the force applied and the distance from the pivot point (lever arm). In this scenario, the torque applied to the inner cylinder is crucial for maintaining its rotational speed against the resistance provided by the viscosity of the liquid.

Rotational Speed

Rotational speed refers to how fast an object rotates around an axis, typically measured in revolutions per minute (rev/min). In this question, the inner cylinder's rotational speed is essential for calculating viscosity, as it influences the shear rate experienced by the liquid between the cylinders, which directly relates to the viscosity measurement.