A 2.0 kg block on a horizontal, frictionless surface is connected by a massless spring and a massless, frictionless pulley to a hanging mass. For what value of the hanging mass does the block accelerate at 1.5 m/s²?

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Identify the forces acting on the system: The horizontal block experiences a tension force from the spring, while the hanging mass experiences both the gravitational force (weight) and the tension force in the opposite direction.

Write Newton's second law for the horizontal block: The net force on the block is the tension in the spring, which causes the block to accelerate. This can be expressed as \( T = m_1 a \), where \( m_1 = 2.0 \ \text{kg} \) and \( a = 1.5 \ \text{m/s}^2 \).

Write Newton's second law for the hanging mass: The net force on the hanging mass is the difference between its weight and the tension in the spring. This can be expressed as \( m_2 g - T = m_2 a \), where \( m_2 \) is the mass of the hanging object, \( g = 9.8 \ \text{m/s}^2 \), and \( a = 1.5 \ \text{m/s}^2 \).

Substitute the expression for tension \( T \) from the first equation into the second equation: Replace \( T \) in \( m_2 g - T = m_2 a \) with \( m_1 a \), resulting in \( m_2 g - m_1 a = m_2 a \).

Solve for \( m_2 \): Rearrange the equation \( m_2 g - m_1 a = m_2 a \) to isolate \( m_2 \). Combine like terms and factor out \( m_2 \), leading to \( m_2 = \frac{m_1 a}{g - a} \). Substitute the known values of \( m_1 \), \( a \), and \( g \) to find the value of the hanging mass.

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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 acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed by the formula F = ma, where F is the net force, m is the mass, and a is the acceleration. In this scenario, understanding how the forces acting on both the block and the hanging mass relate to their respective accelerations is crucial.

Spring Force

The force exerted by a spring is described by Hooke's Law, which states that the force is proportional to the displacement of the spring from its equilibrium position. This is expressed as F_s = -kx, where F_s is the spring force, k is the spring constant, and x is the displacement. In this problem, the spring's force will play a significant role in determining the net force acting on the block.

Free Body Diagram

A free body diagram (FBD) is a graphical representation used to visualize the forces acting on an object. It helps in identifying all the forces, including tension, gravitational force, and spring force, acting on the block and the hanging mass. By analyzing the FBDs for both masses, one can apply Newton's laws to solve for the unknown hanging mass that results in the specified acceleration.