A 50 kg box hangs from a rope. What is the tension in the rope if: The box has v_y = 5.0 m/s and is speeding up at 5.0 m/s²

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Identify the forces acting on the box: The box is subject to two forces: the gravitational force (weight) acting downward and the tension in the rope acting upward. The net force is responsible for the box's acceleration.

Write the equation for the net force using Newton's Second Law: \( F_{\text{net}} = ma \), where \( m \) is the mass of the box and \( a \) is its acceleration. The net force is also the difference between the tension \( T \) and the gravitational force \( F_g \).

Express the gravitational force: \( F_g = mg \), where \( g \) is the acceleration due to gravity (approximately \( 9.8 \ \text{m/s}^2 \)). Substitute \( m = 50 \ \text{kg} \) into this equation.

Combine the equations: The net force is \( T - F_g = ma \). Rearrange this to solve for the tension: \( T = ma + F_g \). Substitute \( F_g = mg \) into the equation, resulting in \( T = ma + mg \).

Substitute the known values: Use \( m = 50 \ \text{kg} \), \( a = 5.0 \ \text{m/s}^2 \), and \( g = 9.8 \ \text{m/s}^2 \) into the equation \( T = ma + mg \). Simplify to find the tension in the rope.

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Newton's Second Law of Motion

Newton's Second Law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration (F = ma). This principle is crucial for analyzing the forces acting on the box, including gravitational force and tension in the rope, especially when the box is accelerating.

Tension in a Rope

Tension is the force transmitted through a rope or string when it is pulled tight by forces acting from opposite ends. In this scenario, the tension in the rope must counteract both the weight of the box and provide the additional force required for its upward acceleration.

Weight of an Object

The weight of an object is the force exerted on it due to gravity, calculated as the product of its mass and the acceleration due to gravity (W = mg). For the 50 kg box, this weight must be considered alongside the upward tension when determining the net force acting on the box as it accelerates.