Ch 06: Dynamics I: Motion Along a Line

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 06: Dynamics I: Motion Along a Line

Problem 58

Chapter 6, Problem 58

The 2.0 kg wood box in FIGURE P6.58 slides down a vertical wood wall while you push on it at a 45° angle. What magnitude of force should you apply to cause the box to slide down at a constant speed?

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Identify the forces acting on the box: (1) the gravitational force $ F_g = m \cdot g $, where $ m = 2.0 \ \text{kg} $ and $ g = 9.8 \ \text{m/s}^2 $, (2) the normal force $ F_N $ due to your applied force, (3) the frictional force $ F_f $, and (4) the applied force $ F_{app} $ at a 45° angle.

Break the applied force $ F_{app} $ into components: (1) the horizontal component $ F_{app,x} = F_{app} \cdot \cos(45°) $, which contributes to the normal force, and (2) the vertical component $ F_{app,y} = F_{app} \cdot \sin(45°) $, which opposes the gravitational force.

Write the condition for constant speed: Since the box slides at a constant speed, the net force in both the vertical and horizontal directions must be zero. This means the downward force (gravity) is balanced by the upward forces (friction and the vertical component of the applied force).

Express the frictional force: The frictional force is given by $ F_f = \mu \cdot F_N $, where $ \mu $ is the coefficient of kinetic friction for wood on wood (typically $ \mu \approx 0.2 $), and $ F_N $ is the normal force. The normal force is equal to the horizontal component of the applied force, $ F_N = F_{app,x} = F_{app} \cdot \cos(45°) $.

Set up the force balance equations: (1) In the vertical direction, $ F_g = F_f + F_{app,y} $, or $ m \cdot g = \mu \cdot (F_{app} \cdot \cos(45°)) + F_{app} \cdot \sin(45°) $. Solve this equation for $ F_{app} $ to find the magnitude of the applied force needed to maintain constant speed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton's First Law of Motion

Newton's First Law states that an object at rest will remain at rest, and an object in motion will continue in motion at a constant velocity unless acted upon by a net external force. In this scenario, for the box to slide down at a constant speed, the net force acting on it must be zero, meaning the applied force must balance the forces of gravity and friction.

Friction

Friction is the resistive force that opposes the motion of an object sliding over a surface. It depends on the nature of the surfaces in contact and the normal force acting between them. In this case, the frictional force will play a crucial role in determining the magnitude of the applied force needed to maintain constant speed as the box slides down the wall.

Components of Force

When a force is applied at an angle, it can be resolved into horizontal and vertical components using trigonometric functions. For the 45° angle in this problem, both the horizontal and vertical components of the applied force will affect the box's motion. Understanding how to calculate these components is essential for determining the correct magnitude of the force needed to achieve constant speed.

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