A 2.80-kg grinding wheel is in the form of a solid cylinder of radius 0.100 m. (a) What constant torque will bring it from rest to an angular speed of 1200 rev/min in 2.5 s?

Verified step by step guidance

1. Calculate the angular speed in radians per second. Since the angular speed is given in revolutions per minute (rev/min), convert it to radians per second using the conversion factor: $1 \text{ rev} = 2\pi \text{ radians}$ and $1 \text{ minute} = 60 \text{ seconds}$. The formula to use is $\omega = \text{angular speed in rev/min} \times \frac{2\pi}{60}$.
2. Determine the angular acceleration, $\alpha$, using the formula $\alpha = \frac{\Delta \omega}{\Delta t}$, where $\Delta \omega$ is the change in angular velocity and $\Delta t$ is the time interval over which this change occurs.
3. Use the moment of inertia for a solid cylinder, $I = \frac{1}{2} m r^2$, where $m$ is the mass of the cylinder and $r$ is its radius, to calculate the moment of inertia of the grinding wheel.
4. Calculate the torque required using the formula $\tau = I \alpha$, where $\tau$ is the torque, $I$ is the moment of inertia calculated in the previous step, and $\alpha$ is the angular acceleration.
5. Ensure all units are consistent when performing calculations (e.g., mass in kilograms, radius in meters, time in seconds) to obtain the correct value for torque in Newton-meters.

Key Concepts

Torque

Torque is a measure of the rotational force applied to an object, causing it to rotate about an axis. It is calculated as the product of the force applied and the distance from the axis of rotation (lever arm). In this context, torque is essential for determining how much force is needed to accelerate the grinding wheel to a specified angular speed.

Moment of Inertia

Moment of inertia is a property of a body that quantifies its resistance to angular acceleration about an axis. For a solid cylinder, it is calculated using the formula I = (1/2) m r², where m is the mass and r is the radius. Understanding moment of inertia is crucial for calculating the angular acceleration of the grinding wheel when torque is applied.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically measured in radians per second squared (rad/s²). It can be calculated using the formula α = Δω / Δt, where Δω is the change in angular velocity and Δt is the time interval. In this problem, determining the angular acceleration is necessary to find the required torque to achieve the desired speed.