Ch 09: Work and Kinetic Energy

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 09: Work and Kinetic Energy

Problem 45

Chapter 9, Problem 45

A 150 g particle at x = 0 is moving at 2.00 m/s in the + x - direction. As it moves, it experiences a force given by Fₓ = (0.250 N) sin (x/2.00 m). What is the particle's speed when it reaches x = 3.14 m?

Verified step by step guidance

Step 1: Recognize that the work-energy principle applies here. The work done by the force on the particle will result in a change in its kinetic energy. The work-energy theorem states: ΔK = W, where ΔK is the change in kinetic energy and W is the work done by the force.

Step 2: Write the expression for kinetic energy. The kinetic energy of the particle is given by K = (1/2)mv², where m is the mass of the particle and v is its velocity. The change in kinetic energy is ΔK = (1/2)m(v_f² - v_i²), where v_f is the final velocity and v_i is the initial velocity.

Step 3: Calculate the work done by the force. The work done by a variable force is given by W = ∫ Fₓ dx, where Fₓ is the force as a function of position x. Substitute Fₓ = (0.250 N) sin(x / 2.00 m) into the integral: W = ∫ (0.250 N) sin(x / 2.00 m) dx, with limits of integration from x = 0 to x = 3.14 m.

Step 4: Solve the integral for work. The integral of sin(ax) is (-1/a) cos(ax), where a is a constant. Here, a = 1 / (2.00 m). Perform the integration and evaluate the result at the limits x = 0 and x = 3.14 m.

Step 5: Use the work-energy theorem to find the final speed. Substitute the calculated work W and the given mass m = 150 g = 0.150 kg into the equation ΔK = W. Solve for v_f using v_f² = v_i² + (2W / m), where v_i = 2.00 m/s. This will give the final speed of the particle.

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Key Concepts

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

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed mathematically as F = ma, where F is the net force, m is the mass, and a is the acceleration. Understanding this law is crucial for analyzing how the force acting on the particle affects its motion and speed.

Work-Energy Principle

The Work-Energy Principle states that the work done on an object is equal to the change in its kinetic energy. In this scenario, the force acting on the particle does work as it moves from x = 0 to x = 3.14 m, which will change its speed. This principle allows us to relate the force to the particle's speed by calculating the work done over the distance traveled.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 0.5mv², where m is the mass and v is the velocity of the object. As the particle moves and experiences a force, its kinetic energy will change, which can be analyzed to determine its final speed at a given position. Understanding kinetic energy is essential for solving the problem and finding the particle's speed at x = 3.14 m.

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