A 100 g particle experiences the one-dimensional, conservative force F_x shown in FIGURE P10.60. Suppose the particle is shot to the right from x = 1.0 m with a speed of 25 m/s. Where is its turning point?

검증된 단계별 안내

Step 1: Understand the concept of a turning point. A turning point occurs when the particle's kinetic energy is completely converted into potential energy, meaning its velocity becomes zero at that position.

Step 2: Use the principle of conservation of mechanical energy. The total mechanical energy (kinetic energy + potential energy) of the particle remains constant because the force is conservative. Write the equation: \( E_{total} = K + U \), where \( K \) is the kinetic energy and \( U \) is the potential energy.

Step 3: Calculate the initial total mechanical energy. The initial kinetic energy is given by \( K = \frac{1}{2} m v^2 \), where \( m \) is the mass of the particle (convert 100 g to kg: \( m = 0.1 \, \text{kg} \)) and \( v \) is its initial velocity (25 m/s). The initial potential energy \( U \) at \( x = 1.0 \, \text{m} \) can be determined from the force diagram or the potential energy function provided in the problem.

Step 4: Determine the potential energy \( U \) at different positions \( x \) using the relationship between force and potential energy: \( F_x = -\frac{dU}{dx} \). Integrate the force function \( F_x \) to find \( U(x) \), ensuring you account for the constant of integration based on the reference point.

Step 5: Solve for the turning point. At the turning point, the kinetic energy is zero, so \( E_{total} = U(x) \). Set the total mechanical energy equal to the potential energy function \( U(x) \) and solve for \( x \). This will give the position where the particle stops and reverses direction.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Conservative Forces

Conservative forces are forces that do not dissipate energy and depend only on the position of an object. The work done by a conservative force on an object moving between two points is independent of the path taken. Examples include gravitational and elastic forces. In this context, the conservative force Fx will determine the potential energy associated with the particle's position.

Kinetic and Potential Energy

Kinetic energy is the energy of an object due to its motion, calculated as KE = 1/2 mv², where m is mass and v is velocity. Potential energy, on the other hand, is the stored energy based on an object's position in a force field, such as gravitational or elastic potential energy. The conservation of mechanical energy principle states that the total mechanical energy (kinetic + potential) remains constant in a conservative system, allowing us to find the turning point of the particle.

Turning Point

The turning point of a particle in motion is the position where its velocity becomes zero, indicating a change in direction. At this point, all kinetic energy has been converted into potential energy. To find the turning point, one must analyze the energy conservation between kinetic and potential energy, using the initial speed and the potential energy function derived from the conservative force acting on the particle.