Problem 50 - Contemporary Precalculus

Question

An object at the origin is acted upon by two forces, \(u\) and \(v,\) with direction angle \(\theta_{u}\) and \(\theta_{w}\) respectively. Find the direction and magnitude of the resultant force. $$\mathbf{u}=30 \text { newtons, } \theta_{\mathbf{u}}=300^{\circ} ; \mathbf{v}=80 \text { newtons, } \theta_{\mathbf{v}}=40^{\circ}$$

Step-by-Step Solution

Answer

Answer: The magnitude of the resultant force is 52.69 N, and its direction angle is 29.49°.

Step 1: Find the horizontal and vertical components of the forces

To add the forces, we first need to find the horizontal and vertical components of each force. We can use the trigonometric functions cosine and sine to determine these components. For force u, the horizontal component is `u_x = u * cos(θ_u)` and the vertical component is `u_y = u * sin(θ_u)`. Similarly, for force v, the horizontal component is `v_x = v * cos(θ_v)` and the vertical component is `v_y = v * sin(θ_v)`.

Step 2: Calculate the components of the forces

Now we can plug the given magnitudes and direction angles into the equations we found in Step 1. For force u: `u_x = 30 * cos(300°) = -15` `u_y = 30 * sin(300°) = -25.98` For force v: `v_x = 80 * cos(40°) = 61.24` `v_y = 80 * sin(40°) = 51.42`

Step 3: Find the components of the resultant force

To find the horizontal (R_x) and vertical (R_y) components of the resultant force, add the corresponding components of the individual forces: `R_x = u_x + v_x = -15 + 61.24 = 46.24` `R_y = u_y + v_y = -25.98 + 51.42 = 25.44`

Step 4: Calculate the magnitude of the resultant force

To find the magnitude of the resultant force, use the Pythagorean theorem: `R = sqrt(R_x^2 + R_y^2) = sqrt(46.24^2 + 25.44^2) = 52.69 N`

Step 5: Find the direction angle of the resultant force

Finally, to find the direction angle (θ_R) of the resultant force, use the tangent function and the calculated components of the resultant force: `tan(θ_R) = R_y / R_x` `θ_R = arctan(R_y / R_x) = arctan(25.44 / 46.24) = 29.49°` So, the direction and magnitude of the resultant force are 52.69 N and 29.49°.

Key Concepts

Resultant Force

When dealing with forces, it's often essential to determine how they combine to affect an object. This combination of forces is termed the "resultant force". It represents the single force that can replace multiple forces acting on an object, without changing its state of motion. In our exercise, the forces \( \mathbf{u} \) of 30 newtons and \( \mathbf{v} \) of 80 newtons are being combined.

To calculate the resultant force, we first find the components, which allows us to consider the impact of the forces in different directions separately. Then, by using vector addition, we combine these components to get the resultant vector's magnitude and direction.

Here, the magnitude of 52.69 N and direction angle of 29.49° tell us both how strong the force is and the direction in which it's acting.

Trigonometric Functions

To find the components of the forces, trigonometric functions are our best friends. These functions (specifically sine and cosine) relate the angles of a right triangle to its side lengths. For any force represented as a vector, we can break it down into horizontal and vertical components using these functions.

- For force \( \mathbf{u} \), we calculate:
\( u_x = u \, \cos(\theta_u) \)
\( u_y = u \, \sin(\theta_u) \)
- Similarly, for force \( \mathbf{v} \), we have:
\( v_x = v \, \cos(\theta_v) \)
\( v_y = v \, \sin(\theta_v) \)

In simple terms, cosine helps find the adjacent side or horizontal component, while sine provides the opposite side or vertical component. So by knowing the force's direction angle, trigonometry helps in expressing these vector forces in a simpler, more manageable form.

Direction Angles

Direction angles are crucial in understanding how forces act in different orientations in space. They tell us specifically which "way" a force is pointing. In the problem, each force has its direction angle: \( \theta_{\mathbf{u}} = 300^{\circ} \) and \( \theta_{\mathbf{v}} = 40^{\circ} \). These angles are measured from the positive x-axis in standard position.

Knowing the direction angle is key to using trig functions to find components. However, direction angles can sometimes be in degrees or radians, and handling them correctly is vital for accurate calculations. Keep in mind:

- A 300° angle, like with \( \theta_{\mathbf{u}} \), indicates a direction in the fourth quadrant of the coordinate plane.
- A 40° angle situates \( \theta_{\mathbf{v}} \) in the first quadrant.

Thus, direction angles not only inform the magnitude calculations but also ensure that force components reflect the correct orientation.

Force Components

Force components break down the actions of a force into vertical and horizontal influences. This makes analyzing forces in a plane both possible and practical. By focusing separately on these components, understanding the overall effect becomes straightforward.

- For \( \mathbf{u} \):
- Horizontal Component: \( u_x = -15 \)
- Vertical Component: \( u_y = -25.98 \)

- For \( \mathbf{v} \):
- Horizontal Component: \( v_x = 61.24 \)
- Vertical Component: \( v_y = 51.42 \)

The components directly influence the calculation of the resultant force. By adding the horizontal parts together and the vertical ones separately, we form resultant components \( R_x \) and \( R_y \). These components offer a clearer picture of the force's action by reducing complexities related to forces interacting in different planes. Physics becomes manageable by transforming all actions into perpendicular components.