Problem 39 - College Physics

Question

A woman takes her dog Rover for a walk on a leash. To get the little dog moving forward, she pulls on the leash with a force of \(20.0 \mathrm{~N}\) at an angle of \(37^{\circ}\) above the horizontal. (a) How much force is tending to pull Rover forward? (b) How much force is tending to lift Rover off the ground?

Step-by-Step Solution

Answer

(a) Forward force: 15.97 N, (b) Lifting force: 12.036 N.

Step 1: Identify the Forces at Play

The woman is pulling the leash with a force of \( F = 20.0 \text{ N} \) at an angle of \( 37^{\circ} \) above the horizontal. We need to resolve this force into horizontal and vertical components using trigonometry.

Step 2: Calculate Horizontal Force (Forward Component)

The horizontal component of the force, which tends to pull Rover forward, can be calculated using the cosine function: \[ F_{\text{horizontal}} = F \cdot \cos(37^{\circ}) \]Substituting the values, we get: \[ F_{\text{horizontal}} = 20.0 \cdot \cos(37^{\circ}) \approx 20.0 \cdot 0.7986 = 15.97 \text{ N} \]

Step 3: Calculate Vertical Force (Lifting Component)

The vertical component of the force, which tends to lift Rover off the ground, can be calculated using the sine function: \[ F_{\text{vertical}} = F \cdot \sin(37^{\circ}) \]Substituting the values, we get: \[ F_{\text{vertical}} = 20.0 \cdot \sin(37^{\circ}) \approx 20.0 \cdot 0.6018 = 12.036 \text{ N} \]

Key Concepts

Trigonometry in Physics

Trigonometry is a powerful mathematical tool that helps us understand various forces in physics. It allows us to break down a force that acts at an angle into two perpendicular components: horizontal and vertical.
When a force is applied at an angle, it can be decomposed using trigonometric functions, sine and cosine. In our scenario, a force of 20 N is exerted at a 37-degree angle. The components, horizontal and vertical, describe the force's effect in those directions.

• The **cosine** of the angle helps find the horizontal component (force parallel to the ground). This is the part of the force that moves Rover forward.
• The **sine** of the angle is used for the vertical component (force perpendicular to the ground). This part potentially lifts Rover off the ground.

Breaking down forces is critical because it enables us to analyze their specific effects separately.

Force Components

Force in physics can often be complex, acting in different directions simultaneously. Thus, resolving a force into its components is fundamental to understand its true impact. Each force vector can be divided into two main components

• The **horizontal component** is found using the formula: \[ F_{\text{horizontal}} = F \cdot \cos(\theta) \]These calculations determine how much of the force actually contributes to forward motion. For example, the horizontal component of the 20 N force is \[ 20.0 \cdot \cos(37^{\circ}) \approx 15.97 \text{ N} \]
• The **vertical component** is calculated with:\[ F_{\text{vertical}} = F \cdot \sin(\theta) \]This identifies how much of the force acts to lift an object upwards; in this case, about \[ 20.0 \cdot \sin(37^{\circ}) \approx 12.036 \text{ N} \]

Understanding these components helps predict and calculate physical movements effectively.

Newton's Laws of Motion

Newton's Laws of Motion give us a foundational framework for understanding how objects move. They are essential for analyzing any scenario involving forces.

• **First Law**: An object will remain at rest, or in uniform motion in a straight line, unless acted upon by a force. The force applied by the woman initiates Rover's movement.
• **Second Law**: The acceleration of an object is dependent on the net forces acting upon it and its mass, described by the equation \[ F = m \cdot a \]If Rover's mass and acceleration are known, we can further understand how he'll move with the applied force.
• **Third Law**: For every action, there's an equal and opposite reaction. When the woman pulls on the leash, there's a force exerted back on her in the opposite direction.

Applying these laws to the exercise not only helps solve for the force components, but also ensures a deeper understanding of the interaction between Rover, the engaging forces, and the resulting motion.