91Ó°ÊÓ

In physics, work is a concept that involves a force causing an object to move. Work is calculated using the formula:

• \[W = F \cdot d \cdot \cos(\theta)\]

Here, \(W\) represents work, \(F\) is the force applied, \(d\) is the distance over which the force acts, and \(\theta\) is the angle between the force and the direction of motion.
This equation helps us determine how much energy is transferred when an object moves under the influence of a force.
It's crucial to understand that work is only done when the movement of an object aligns with the direction of the force applied. For instance, if the force is perpendicular to the motion, like in the case of gravitational force on a horizontal movement, no work is done. Work can be positive when the force's direction aligns with the displacement and negative if opposite, such as with friction.

Force is a push or pull acting on an object, resulting from its interaction with another object. When multiple forces act on an object, the net force determines its motion. In the context of our shopping cart exercise, the shopper's force is crucial to balance out friction and other forces.

We use trigonometry to resolve the direction and magnitude of forces. The force exerted by the shopper is calculated using trigonometric functions:

• The horizontal component is \( F_{x} = F \cdot \cos(29^{\circ}) \).

This component must counteract the frictional force to maintain constant velocity.
Thus, solving for \(F\) involves understanding this relationship and using trigonometry to ensure the proper balancing of forces in the horizontal plane.

Friction is the force that opposes the relative motion or tendency of such motion between two surfaces in contact. It acts opposite to the direction of motion and can make it more challenging to move objects, like the shopping cart in the exercise. The shopping cart is subject to a 48 N frictional force.
The work done by friction is always negative since its force direction opposes displacement. This is calculated using:

• \[W_{friction} = F_{friction} \times d \times \cos(180^{\circ})\]

Here, the use of \(\cos(180^{\circ})\) reflects the opposing nature, leading to negative work. This negative value indicates energy is being used to overcome friction.

Trigonometry plays a fundamental role in breaking down forces into components, especially when forces act at angles. For the shopper pushing the cart at 29 degrees below the horizontal, we employ trigonometric functions to resolve the force accurately.
In our calculations:

• We use \(\cos(\theta)\) to determine the horizontal force component.
• This allows us to relate the applied force to the necessary counterpart that balances friction.

Trigonometry enables precise calculations in physics by addressing the angled force applications and ensuring each component of force is correctly applied in analyses, maintaining clarity and accuracy in problem-solving scenarios.