Problem 49 - Algebra and Trigonometry

Question

Force \(A\) car is on a driveway that is inclined \(25^{\circ}\) to the horizontal. If the car weighs 2755 Ib, find the force required to keep it from rolling down the driveway.

Step-by-Step Solution

Answer

The force required is approximately 1164.56 lb.

Step 1: Understand the Problem

We have a car on an inclined plane, and we need to find the force that opposes its tendency to roll down due to gravity. This force is the component of the gravitational force that acts parallel to the incline.

Step 2: Identify Components of Weight

The weight of the car, given as 2755 lb, acts vertically downwards. For an inclined plane, we need to resolve this into two components: one perpendicular to the plane and one parallel to the plane.

Step 3: Project the Gravitational Force Along the Incline

The component of the gravitational force acting along the incline can be found using the sine function because it is opposite the angle of inclination: \[ F_{\text{parallel}} = W \cdot \sin(\theta) \] where \( W = 2755 \text{ lb} \) and \( \theta = 25^{\circ} \).

Step 4: Calculate the Parallel Component

Substitute the given values into the formula: \[ F_{\text{parallel}} = 2755 \cdot \sin(25^{\circ}) \]Calculate the value using trigonometric values, \[ F_{\text{parallel}} \approx 2755 \cdot 0.4226 \approx 1164.56 \text{ lb} \]

Step 5: Conclusion

The force required to keep the car from rolling down the driveway is approximately 1164.56 lb.

Key Concepts

Gravitational Force

Gravitational force is an important concept when working with inclined planes. It is the force that Earth exerts on any object, pulling it towards the center. For our example, this force is determined by the weight of the car, which is given as 2755 lb. This weight acts vertically downward.

When dealing with inclined planes, this vertical force does not act directly along the surface of the incline. Therefore, it is essential to understand how this force can be split into different directions to analyze it accurately. Specifically, it needs to be resolved into components that align with the slope and normal, or perpendicular, to the surface of the incline.

Trigonometric Functions

Trigonometric functions play a crucial role in resolving forces on an inclined plane. In this context, they help us understand and calculate the components of gravitational force acting on the car.

Trigonometry focuses on the relationships between the angles of a triangle and its sides. The three basic functions are sine, cosine, and tangent. When dealing with inclined planes:

• Sine (\( \sin \)) is used to calculate the force component that runs parallel to the plane.
• Cosine (\( \cos \)) helps in finding the force component perpendicular to the plane.

For our problem, the angle of inclination is 25 degrees. To find the parallel force that needs balancing to prevent the car from rolling, we use the sine function.

Force Resolution

Force resolution is the process of breaking down a single force into perpendicular components to analyze motion. This is particularly useful on inclined planes where the direct effects of gravity are not aligned with the plane's surface.

In the given problem, we resolve the car's weight of 2755 lb into two components: one parallel and one perpendicular to the slope. The parallel component is the focus since it influences the tendency of the car to roll backward.

The calculation involves using the formula\[ F_{\text{parallel}} = W \cdot \sin(\theta) \]where:

• \( W \) is the weight (2755 lb).
• \( \theta \) is the angle of inclination (25 degrees).

By substituting and calculating, we find that the force required to stop the car from rolling is approximately 1164.56 lb, using the sine of 25 degrees.