91Ó°ÊÓ

Torque is the rotational equivalent of force in linear motion. In this problem, we're calculating the torque needed to accelerate a grinding wheel. Torque is essential for causing rotational changes in an object's motion. To find torque, we use the formula:\[ \tau = I \alpha \]Where:

• \( \tau \) is torque
• \( I \) is the moment of inertia
• \( \alpha \) is the angular acceleration

The moment of inertia (\( I \)) is a measure of an object's resistance to changes in its rotation. For a uniform cylinder like the grinding wheel, the moment of inertia is calculated using:\[ I = \frac{1}{2} m r^2 \]This involves the mass (\( m \)) and radius (\( r \)) of the wheel. The angular acceleration (\( \alpha \)) is the rate of change of angular velocity over time. Knowing these values and the formula helps us compute how much torque is required.It's important to remember that the direction of torque can influence how an object rotates. Torque can be adjusted by changing forces or by changing the lever arm's length or direction.

Angular velocity describes how quickly an object rotates and is an important concept in rotational dynamics. It's measured in radians per second (rad/s) and helps us understand how fast something is spinning.A key step in this exercise is converting the wheel's speed from revolutions per minute (rpm) to radians per second. The conversion uses the relation:\[ 1 \text{ rpm} = \frac{2\pi}{60} \text{ rad/s} \]For the grinding wheel starting from rest, the final angular velocity (\( \omega_f \)) is computed by transforming the given speed. This results in \( \omega_f \approx 183.26 \text{ rad/s} \).Understanding angular velocity helps determine how forces like torque affect the wheel's rotation speed. Angular velocity impacts the performance of rotating objects in real-world applications, such as machinery or engines.

Angular acceleration is the rate at which angular velocity changes with time. It provides insight into how quickly an object can ramp up its spinning speed.The formula for angular acceleration is:\[ \alpha = \frac{\omega_f - \omega_i}{t} \]Where:

• \( \alpha \) is angular acceleration
• \( \omega_f \) is final angular velocity
• \( \omega_i \) is initial angular velocity
• \( t \) is the time period

In this exercise, since the wheel starts from rest, the initial angular velocity \( \omega_i \) is 0, hence:\[ \alpha = \frac{183.26 - 0}{5} = 36.652 \text{ rad/s}^2 \]This value indicates how fast the wheel's spinning speed increases. Understanding angular acceleration is crucial for designing systems where the rotational speed needs to change rapidly, ensuring smooth and precise operations in mechanical systems.