91Ó°ÊÓ

Torque is a measure of the rotational force acting on an object. It’s what makes things spin and is calculated using the formula \( \tau = r \times f \). In this case, \( \tau \) represents torque, \( r \) represents the radius at which the force is applied, and \( f \) is the force itself.
In our grinding tool problem, the frictional force \( f \) acting on the tool, calculated as \( f = \mu_k \times F = 0.320 \times 180 \, \text{N} = 57.6 \, \text{N} \), is applied at the rim of the wheel. The radius \( r \) of 0.2 meters means our torque exerted by the frictional force is:

• \( \tau = 0.2 \, \text{m} \times 57.6 \, \text{N} \)
• \( \tau = 11.52 \, \text{Nm} \)

Torque helps us understand how efficiently a force can create rotational motion, which in turn affects power and energy transfer from the grinding wheel.

Angular velocity measures how fast something spins or rotates, expressed in radians per second. In problems involving rotating objects, such as our grinding wheel, converting linear speed into angular speed helps clarify how quickly the object is turning.
For our wheel, rotating at \(2.50 \, \text{rev/s}\), the angular velocity \( \omega \) is found by:

• Using the formula \( \omega = 2\pi \times \text{revolutions per second} \)
• \( \omega = 2\pi \times 2.50 = 5\pi \, \text{rad/s} \)

Understanding angular velocity is crucial for analyzing the power of rotating systems, as it directly influences how much work can be done over time in systems like our grinding machine.

Power transfer in mechanical systems refers to the rate at which energy is transferred from one part of a system to another. In this context, it involves moving energy from the motor to the grinding wheel, where it becomes thermal energy and kinetic energy.
To find the power being transferred, we use the formula \( P = \tau \times \omega \), where \( P \) is power, \( \tau \) is torque, and \( \omega \) is the angular velocity.
Substituting the values from our exercise:

• Torque \( \tau = 11.52 \, \text{Nm} \)
• Angular velocity \( \omega = 5\pi \, \text{rad/s} \)
• Power \( P = 11.52 \, \text{Nm} \times 5\pi \, \text{rad/s} = 180\pi \, \text{W} \)
• Approximately \( 565.5 \, \text{W} \)

This power represents how efficiently energy is being converted from electrical to mechanized energy, effectively making the grinding process work on the tool and improve its sharpness.