Problem 27 - Applied Physics

Question

An ag mechanic tightens implement bolts using \(52.5 \mathrm{~N} \mathrm{~m}\) of torque at a rate of \(2.25 \mathrm{rad} / \mathrm{s}\). What power does the mechanic develop in tightening the bolts?

Step-by-Step Solution

Answer

The power developed is 118.125 W.

Step 1: Understanding the Formula for Power

Power (P) in rotational motion is given by the formula \( P = \tau \cdot \omega \), where \( \tau \) is the torque and \( \omega \) is the angular velocity. In this problem, \( \tau = 52.5 \mathrm{~N} \cdot \mathrm{m} \) and \( \omega = 2.25 \mathrm{~rad/s} \).

Step 2: Substitution into the Power Formula

Substitute the given values into the formula: \( P = 52.5 \mathrm{~N} \cdot \mathrm{m} \cdot 2.25 \mathrm{~rad/s} \).

Step 3: Perform the Calculation

Calculate the power using the substituted values: \( P = 52.5 \times 2.25 = 118.125 \mathrm{~W} \).

Step 4: Conclusion of Solution

The power developed by the mechanic in tightening the bolts is \( 118.125 \mathrm{~W} \).

Key Concepts

Torque

Think of torque as the rotational equivalent of force. When you push or pull an object to make it move straight, you are applying force. But, if you twist or turn an object, you're applying torque. Torque is typically measured in Newton-meters (Nm). This measurement reflects how much rotational force is applied and at what distance from the axis of rotation. In our exercise, the mechanic applies a torque of 52.5 Nm to tighten the bolts. The larger the torque, the more "twisting" power is applied. This is crucial in ensuring that bolts are tightened securely
without slippage. There are a few factors that affect torque:

• Amount of force applied.
• Distance from the axis of rotation.
• Angle at which the force is applied.

Understanding torque is essential in engineering and mechanics because it ensures that components, such as bolts, are fastened with the right amount of force to prevent loosening.

Angular Velocity

Angular velocity is all about how quickly something is spinning. Just like regular velocity tells us how fast something is moving in a straight line, angular velocity tells us the speed of rotation. It's measured in radians per second (rad/s), which indicates how many "radians" an object turns in one second. Radians are useful in circular motion because they relate directly to the circle's radius. In our example, the mechanic tightens the bolts with an angular velocity of 2.25 rad/s. This means the bolt is turning at a consistent rate, completing 2.25 radians every second. Rapid tightening or loosening is mostly dependent on the angular velocity. If you imagine a record on a turntable, angular velocity would determine how quickly it spins.
To sum up:

• Angular velocity conveys rotational speed.
• Measured in radians per second.
• Crucial for understanding rotational dynamics.

This concept becomes especially important when calculating rotational energy or designing mechanical systems that require precise timing.

Power in Physics

Power, in the context of physics, is the rate at which work is done or energy is transferred. In rotational systems, power can also be understood as how fast energy is consumed or produced by turning an object. The formula to calculate power in rotational motion is given by multiplying torque by angular velocity, as shown in the original exercise: \[P = \tau \cdot \omega\]Where:

• \( P \) is the power (in watts).
• \( \tau \) is the torque (in Nm).
• \( \omega \) is the angular velocity (in rad/s).

In the exercise, substituting the provided values yields a power of 118.125 W. This indicates how much energy the mechanic uses per second to tighten the bolts.
Understanding power in this way helps in identifying efficiency and capability in machines and mechanical tasks. Whether it's tightening bolts or running an electric motor, knowing the power involved allows for better design and energy management.