Ch. 08 - Conservation of Energy

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 08 - Conservation of Energy

Problem 69c

Chapter 8, Problem 69c

The position of a 280-g object is given (in meters) by x = 4.0t³ - 8.0t² - 44t, where t is in seconds. What is the average net power input during the interval from t = 0s to t = 2.0 s, and in the interval from t = 2.0 s to 4.0 s?

Verified step by step guidance

Step 1: Understand the problem. The position of the object is given as a function of time: x(t) = 4.0t³ - 8.0t² - 44t (in meters). The goal is to calculate the average net power input during two time intervals: t = 0s to t = 2.0s, and t = 2.0s to t = 4.0s. Power is related to the rate of work done, which depends on force and velocity.

Step 2: Calculate the velocity as a function of time by differentiating the position function x(t) with respect to time t. Use the formula: v(t) = dx/dt. Differentiate x(t) = 4.0t³ - 8.0t² - 44t to get v(t).

Step 3: Calculate the acceleration as a function of time by differentiating the velocity function v(t) with respect to time t. Use the formula: a(t) = dv/dt. Differentiate the velocity function obtained in Step 2 to get a(t).

Step 4: Determine the force acting on the object as a function of time using Newton's second law: F(t) = m * a(t), where m is the mass of the object (280 g = 0.280 kg). Substitute the acceleration function a(t) into this equation to find F(t).

Step 5: Calculate the instantaneous power as a function of time using the formula: P(t) = F(t) * v(t). Then, find the average power over each time interval by integrating P(t) over the interval and dividing by the duration of the interval. Use the formula: P_avg = (1/Δt) * ∫[t₁ to t₂] P(t) dt, where Δt = t₂ - t₁. Perform this calculation for both intervals: t = 0s to t = 2.0s, and t = 2.0s to t = 4.0s.

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Key Concepts

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

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as position, velocity, and acceleration, which are essential for analyzing the motion of the 280-g object described by the position function x(t). Understanding kinematics allows us to derive the object's velocity and acceleration from its position function.

Power

Power is defined as the rate at which work is done or energy is transferred over time. In the context of physics, average power can be calculated by dividing the work done by the time interval during which the work is performed. For the given problem, calculating the average net power input requires determining the work done on the object over the specified time intervals and dividing it by the duration of those intervals.

Work-Energy Principle

The Work-Energy Principle states that the work done on an object is equal to the change in its kinetic energy. This principle is crucial for solving problems involving power, as it allows us to relate the net work done on the object to its motion. By applying this principle, we can calculate the work done during the specified time intervals and subsequently determine the average net power input.

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The graph of Fig. 8–43 shows the potential energy curve of a particle moving along the 𝓍 axis under the influence of a conservative force. Note that the total energy E > U(𝓍), so that the particle’s speed is never zero. At what value(s) of 𝓍 is the magnitude of the force a minimum?

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