How To Find Applied Force

How to Find Applied Force: A Comprehensive Guide

Finding the applied force might seem straightforward, but the approach varies significantly depending on the context. This comprehensive guide will explore various methods for determining applied force, from simple scenarios involving Newton's second law to more complex situations involving energy considerations, friction, and inclined planes. We'll delve into the theoretical underpinnings, practical applications, and common pitfalls to ensure you grasp the concept thoroughly. Understanding how to find applied force is crucial in various fields, from basic physics to advanced engineering, and this guide aims to equip you with the necessary knowledge.

Understanding Force and its Measurement

Before diving into the methods, let's establish a firm understanding of what force is. In physics, force is an interaction that, when unopposed, will change the motion of an object. This change can be a change in speed, direction, or both. Force is a vector quantity, meaning it has both magnitude (size) and direction. The SI unit for force is the newton (N), defined as 1 kg⋅m/s².

Measuring force directly can be challenging in many scenarios. Instead, we often infer the applied force by observing its effects on an object, usually through changes in its motion or through the use of specialized instruments like force gauges or load cells. These instruments measure the force exerted on them, allowing indirect determination of the applied force.

Method 1: Using Newton's Second Law of Motion (F=ma)

This is the most fundamental and widely applicable method for finding applied force. Newton's second law states that the net force acting on an object is equal to the product of its mass and acceleration. Mathematically, this is represented as:

F = ma

Where:

• F represents the net force (in Newtons)
• m represents the mass of the object (in kilograms)
• a represents the acceleration of the object (in meters per second squared)

Applying Newton's Second Law:

To find the applied force using this method, you need to know the mass and acceleration of the object. Let's consider a few examples:

• Example 1: Constant Acceleration: A 10 kg box is pushed across a frictionless surface, accelerating at 2 m/s². The applied force is: F = (10 kg)(2 m/s²) = 20 N. The force is in the direction of the acceleration.

• Example 2: Multiple Forces: A 5 kg block is pulled horizontally with a force of 25 N, while experiencing a frictional force of 5 N in the opposite direction. The net force is 25 N - 5 N = 20 N. Using F=ma, we can determine the acceleration: a = F/m = 20 N / 5 kg = 4 m/s².

• Example 3: Inclined Plane: A 2 kg object slides down a frictionless inclined plane with an acceleration of 4 m/s². The force causing this acceleration is the component of gravity acting parallel to the plane. To find this force, we multiply the mass by the acceleration: F = (2 kg)(4 m/s²) = 8 N. This example demonstrates that the applied force doesn't always need to be a direct push or pull; it can be a component of another force.

Limitations:

This method relies on the assumption that the net force is known or can be easily calculated. In situations with complex interactions or multiple unknown forces, this approach might not be sufficient.

Method 2: Using Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. This theorem provides an alternative approach to finding applied force, particularly in situations involving displacement and changes in speed.

W = ΔKE

Where:

• W represents the net work done (in Joules)
• ΔKE represents the change in kinetic energy (in Joules) ΔKE = ½mv² - ½mu², where v and u are final and initial velocities respectively.

Applying the Work-Energy Theorem:

Work is defined as the dot product of force and displacement: W = Fd cosθ, where:

• F is the applied force
• d is the displacement
• θ is the angle between the force and displacement vectors.

If the force is parallel to the displacement (θ = 0°), the equation simplifies to W = Fd.

Example: A 5 kg object is pushed horizontally across a frictionless surface by a constant force, accelerating from rest to 10 m/s over a distance of 20 m. To find the applied force, we first calculate the change in kinetic energy:

ΔKE = ½(5 kg)(10 m/s)² - ½(5 kg)(0 m/s)² = 250 J

Then, using the work-energy theorem:

250 J = F(20 m)

F = 250 J / 20 m = 12.5 N

Limitations:

This method requires knowledge of the displacement and change in kinetic energy. It is less useful in situations where the force is not constant or the displacement is not known.

Method 3: Considering Friction

Friction is a force that opposes motion between surfaces in contact. The force of friction depends on the coefficient of friction (μ) and the normal force (N). The equation for frictional force is:

F_friction = μN

Where:

• F_friction is the frictional force
• μ is the coefficient of friction (static or kinetic)
• N is the normal force

Applying Friction to Find Applied Force:

When dealing with friction, the applied force must overcome the frictional force to initiate or maintain motion. The net force is the difference between the applied force and the frictional force.

Example: A 10 kg block is pushed across a rough surface with a coefficient of kinetic friction of 0.2. The normal force is equal to the weight of the block (mg = 10 kg * 9.8 m/s² = 98 N). If the block accelerates at 1 m/s², the applied force can be found using Newton's second law, considering the frictional force:

F_applied - F_friction = ma

F_applied - (0.2)(98 N) = (10 kg)(1 m/s²)

F_applied = 10 N + 19.6 N = 29.6 N

This example shows that the applied force needs to be larger than the frictional force to cause acceleration.

Method 4: Analyzing Forces on Inclined Planes

Objects on inclined planes experience forces due to gravity and potentially friction. To find the applied force, you need to resolve the gravitational force into components parallel and perpendicular to the plane.

The component of gravity parallel to the plane (F_parallel) is given by:

F_parallel = mg sinθ

Where:

• m is the mass of the object
• g is the acceleration due to gravity
• θ is the angle of inclination

The component of gravity perpendicular to the plane (F_{perpendicular}) is given by:

F_{perpendicular} = mg cosθ

Example: A 5 kg block rests on an inclined plane with an angle of 30°. The coefficient of static friction is 0.3. To find the minimum applied force needed to prevent the block from sliding down, we first calculate the parallel component of gravity:

F_parallel = (5 kg)(9.8 m/s²) sin(30°) ≈ 24.5 N

Then, calculate the maximum static frictional force:

F_{friction (max)} = μN = μmg cosθ = (0.3)(5 kg)(9.8 m/s²) cos(30°) ≈ 12.7 N

Since the parallel component of gravity exceeds the maximum static friction, an applied force is needed to keep the block from sliding. The minimum applied force required would be:

F_{applied (min)} = F_parallel - F_{friction (max)} ≈ 24.5 N - 12.7 N ≈ 11.8 N

Frequently Asked Questions (FAQ)

Q1: What if the applied force is not constant?

If the force varies over time, you'll need to use calculus to determine the net force and acceleration. Newton's second law still applies, but the force and acceleration become functions of time.

Q2: How do I deal with forces in more than one dimension?

Resolve the forces into their component vectors (usually x and y components). Apply Newton's second law separately to each component to find the net force in each direction. Then use vector addition to find the resultant force.

Q3: What are some common errors when finding applied force?

• Neglecting friction: Friction is a significant factor in many real-world scenarios and should not be ignored.
• Incorrectly resolving forces: When dealing with inclined planes or other complex situations, ensure that you accurately resolve forces into their components.
• Confusing mass and weight: Mass and weight are different quantities. Mass is a measure of inertia, while weight is the force due to gravity.
• Units inconsistencies: Always use consistent units (e.g., kilograms, meters, seconds) throughout your calculations.

Conclusion

Finding the applied force requires a thorough understanding of fundamental physics principles, including Newton's laws of motion, the work-energy theorem, and frictional forces. The approach varies depending on the specific scenario. Remember to consider all forces acting on the object, resolve forces into components if necessary, and pay close attention to units to ensure accurate calculations. By mastering these techniques, you'll be well-equipped to solve a wide range of problems involving applied force in various fields of study and applications. This understanding will be invaluable in further exploration of more advanced concepts in physics and engineering.