Question

Two workers pull horizontally on a heavy box, but one pulls twice as hard as the other. The larger pull is directed at 21.0\(^{\circ}\) west of north, and the resultant of these two pulls is 460.0 N directly northward. Use vector components to find the magnitude of each of these pulls and the direction of the smaller pull.

Short answer

Small pull: 193.2 N east of north; large pull: 386.4 N west of north.

Step-by-step solution

Identify forces and angles

We have two forces, \( F_1 \) and \( F_2 \). Let's say \( F_2 = 2F_1 \). The largest force, \( F_2 \), is applied at an angle \( 21.0^{\circ} \) west of north. The resultant force \( R = 460.0 \text{ N}\) is directed due north.

Resolve components of \( F_2 \)

Since \( F_2 \) is directed at \( 21.0^{\circ} \) west of north, its components are:- Northward: \( F_{2y} = F_2 \cos(21.0^{\circ}) \)- Westward: \( F_{2x} = -F_2 \sin(21.0^{\circ}) \) (assuming west is negative x-direction)

Resolve components of \( F_1 \)

Let the direction of \( F_1 \) make an angle \( \theta \) east of north. Its components are:- Northward: \( F_{1y} = F_1 \cos(\theta) \)- Eastward: \( F_{1x} = F_1 \sin(\theta) \) (assuming east is positive x-direction)

Use resultant force condition

Since the resultant force is 460.0 N northward, the sum of the northward components is equal to 460.0 N, and the sum of the horizontal components is zero:1. \( F_1 \cos(\theta) + F_2 \cos(21.0^{\circ}) = 460.0 \)2. \( F_1 \sin(\theta) = F_2 \sin(21.0^{\circ}) \)

Solve simultaneous equations

Substitute \( F_2 = 2F_1 \) in equations from step 4:1. \( F_1 \cos(\theta) + 2F_1 \cos(21.0^{\circ}) = 460.0 \)2. \( F_1 \sin(\theta) = 2F_1 \sin(21.0^{\circ}) \)From equation 2, divide through by \( F_1 \) (assuming \( F_1 eq 0 \)):\[\sin(\theta) = 2 \sin(21.0^{\circ})\]Calculate \( \theta \); \( \theta = \arcsin(2 \sin(21.0^{\circ})) \). Calculate \( \theta \) first.

Find \( F_1 \) using equation 1

Once \( \theta \) is known, substitute into equation 1:\[F_1 (\cos(\theta) + 2 \cos(21.0^{\circ})) = 460.0\]Solve for \( F_1 \).

Calculate \( F_2 \)

Once \( F_1 \) is found, calculate \( F_2 \) using the relationship \( F_2 = 2F_1 \).

Conclusion Step: Summarize the results

\( F_1 = 193.2 \text{ N} \), \( F_2 = 386.4 \text{ N} \), \( \theta = 42.4^{\circ} \) east of north.

Key concepts

Resultant Force

In physics, the resultant force is the single force that represents the vector sum of two or more forces acting on an object. When determining the effect of multiple forces on a single point or object, it's essential to understand how to compute this resultant force. It provides a clearer picture of the net effect all the forces have when acting together.
For instance, if two forces are pulling on an object from different directions, the resultant force is what you get when you "add" these forces together. This combining of forces is done using vector addition, where both the magnitude and direction of the forces are taken into account.
This net force is crucial because it dictates the overall motion of the object according to Newton's laws. In the original exercise, the vectors from two forces are combined to give a resultant force of 460.0 N pointing directly north. This net force direction simplifies the motion analysis because it only needs a single force magnitude and direction instead of dealing with two separate forces.

Trigonometry in Physics

Trigonometry is a powerful mathematical tool used to analyze situations involving angles and distances. It is especially helpful in physics when dealing with vector quantities that have directional components.
When forces are not aligned with standard coordinate axes, trigonometry lets us break these forces into components that are easier to work with. For a force acting at an angle, you can split it into horizontal and vertical components using sine and cosine functions.

• The cosine of the angle gives the component along the axis the angle is measured from. In the given problem, this would be the northward direction for force vectors.
• The sine of the angle gives the component perpendicular to that axis.
• Using these trigonometric relationships allows for a straightforward way to express forces analytically, making it possible to solve for unknown magnitudes or angles.

So in our exercise, trigonometry helps in figuring out how much each force actually contributes to the northward and eastward components by considering their respective angles. This makes solving for unknown vectors possible, as we've separated all forces into manageable northward and lateral parts.

Newton's Laws of Motion

Newton's laws of motion lay the groundwork for understanding how forces affect objects. Each of the three laws provides unique insights into mechanical interactions.
The first law, often called the law of inertia, states that an object at rest stays at rest, and an object in motion stays in motion at constant velocity unless acted upon by a net external force. This principle is subtle in our problem where the combined force keeps the box moving northward but stable in other aspects.
The second law is more directly applied in problems involving forces. It explains that the acceleration of an object is proportional to the net force acting upon it and inversely proportional to its mass with the formula: \( F = ma \).
This law is essential in vector problems since it helps to determine how much force is required to produce a particular acceleration when the system isn't at equilibrium.
The third law states that every action has an equal and opposite reaction. Although not explicitly used in this problem, it is fundamental in the understanding of force interactions. When analyzing the forces exerted by workers on the box, it indirectly underpins the resultant forces computed.
Understanding these laws helps further clarify why the resultant direction is significant and how exactly the box will respond to the combined forces in motion.