Problem 40 - APEX Calculus

Question

A bicycle rider applies \(150 \mathrm{lb}\) of force, straight down, onto a pedal that extends 7in from the crankshaft, making a \(30^{\circ}\) angle with the horizontal. Find the magnitude of the torque applied to the crankshaft.

Step-by-Step Solution

Answer

The magnitude of the torque applied to the crankshaft is 525 inch-pounds.

Step 1: Understanding Torque

Torque, also known as the moment of force, is the measure of the force causing an object to rotate about an axis. It is calculated as the product of the force and the distance from the pivot point to where the force is applied, perpendicular to the lever arm.

Step 2: Torque Formula

The formula for torque \( \tau \) is given by:\[ \tau = r \cdot F \cdot \sin(\theta) \]where:- \( r \) is the distance from the pivot (here, 7 inches),- \( F \) is the force applied (150 lb),- \( \theta \) is the angle between the force direction and the lever arm (30° in this case).

Step 3: Substitute and Calculate

Substitute the given values into the torque formula:\[ \tau = 7 \times 150 \times \sin(30^\circ) \]- \( \sin(30^\circ) = 0.5 \)Now calculate:\[ \tau = 7 \times 150 \times 0.5 \]

Step 4: Compute the Result

Perform the multiplication to find the torque:\[ \tau = 525 \text{ inch-pounds} \]

Key Concepts

Moment of Force

Imagine you're pushing open a door. The farther from the hinge you push, the easier it gets. This is a simple example of the concept known as the moment of force. Simply put, moment of force, or torque, is the effect of a force causing an object to rotate around a pivot or fulcrum. Think of it as the rotational equivalent of linear force. The greater the moment, the more the force will cause the object to rotate. It's important to note that the distance from the pivot point plays a crucial role. The longer the distance, the more leverage you have. In our bicycle example, the pedal acts as the lever arm extending from the crankshaft, which is the pivot, and the force applied by the cyclist creates a rotational motion relative to the crankshaft. This principle underlies many tools and mechanical systems, such as wrenches and see-saws.

Torque Formula

Calculating torque is vital anytime rotation matters, from opening jar lids to tightening bolts. The torque formula helps determine how effectively a force causes rotation. It's expressed as:\[ \tau = r \cdot F \cdot \sin(\theta) \]This formula considers:

• The lever arm length \( r \) (distance from the pivot point to where the force acts)
• The force \( F \) applied
• The angle \( \theta \) between the force vector and the arm

The factor \( \sin(\theta) \) is crucial, as it accounts for only the component of force that's perpendicular to the lever arm. This means only the perpendicular force influences rotation.In most practical situations, torque calculations guide ensuring safety and efficiency, like determining how tightly to secure a bolt or the force needed for a steering system.

Rotational Mechanics

Rotational mechanics is the study of objects in a state of rotation. It involves analyzing forces that cause, maintain, or alter rotational motion. Every rotating object undergoes a kind of motion where rotation happens around a fixed axis or pivot. This is a fundamental aspect of everything from rolling balls to spinning wheels. Several factors influence rotational mechanics:

• Moment of inertia - the resistance of the object to changes in its rotational motion
• Angular velocity - the rate of rotation
• Torque - the measure of the force causing rotation

Understanding these factors is essential when analyzing systems like engines, gears, or even simple bicycles. By applying principles of rotational mechanics, engineers can design systems that optimize performance and efficiency, ensuring that forces applied result in desired rotational motion. In bicycles, riders use torque produced by pushing pedals to create the necessary rotational movement to propel forward.