The position of a 2.0 kg mass is given by x = (2t3 - 3t2) where t is in seconds. What is the net horizontal force on the mass at t = 1s?

Ch 06: Dynamics I: Motion Along a Line

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 06: Dynamics I: Motion Along a Line

문제 16b

6장, 문제 16b

The position of a 2.0 kg mass is given by x = (2t³ - 3t²) where t is in seconds. What is the net horizontal force on the mass at t = 1s?

검증된 단계별 안내

Step 1: Understand the problem. The position of the mass is given as a function of time, x(t) = 2t³ - 3t². To find the net horizontal force, we need to calculate the acceleration at t = 1s using Newton's second law, F = ma, where m is the mass and a is the acceleration.

Step 2: Differentiate the position function x(t) = 2t³ - 3t² with respect to time to find the velocity v(t). The velocity is the first derivative of position: v(t) = dx/dt = d/dt(2t³ - 3t²).

Step 3: Differentiate the velocity function v(t) with respect to time to find the acceleration a(t). The acceleration is the second derivative of position: a(t) = dv/dt = d²x/dt². Perform the differentiation to obtain a(t).

Step 4: Substitute t = 1s into the acceleration function a(t) to calculate the acceleration at that specific time.

Step 5: Use Newton's second law, F = ma, to calculate the net horizontal force. Substitute the mass m = 2.0 kg and the acceleration a(t) at t = 1s into the equation to find the force.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Newton's Second Law of Motion

Newton's Second Law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). This principle is fundamental for analyzing motion, as it allows us to relate the forces acting on an object to its resulting acceleration and, consequently, its position over time.

Acceleration

Acceleration is defined as the rate of change of velocity of an object with respect to time. It can be calculated as the second derivative of position with respect to time. In this problem, we need to find the acceleration of the mass by differentiating the position function twice, which will help us determine the net force acting on the mass.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves equations that describe the relationships between position, velocity, and acceleration. In this question, the position function provided allows us to analyze the motion of the mass and derive the necessary quantities to find the net force.