Textbook Question
The forces in FIGURE EX6.9 act on a 2.0 kg object. What are the values of ax and ay, the x- and y-components of the object's acceleration?
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Ch 06: Dynamics I: Motion Along a Line
Chapter 6, Problem 13b
A 50 kg box hangs from a rope. What is the tension in the rope if: The box moves up at a steady 5.0 m/s?
Verified step by step guidance1
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.
Understand the condition of motion: Since the box is moving upward at a steady speed of 5.0 m/s, it is in dynamic equilibrium. This means the net force acting on the box is zero because there is no acceleration.
Express the forces mathematically: The gravitational force is given by \( F_g = m \cdot g \), where \( m \) is the mass of the box (50 kg) and \( g \) is the acceleration due to gravity (approximately 9.8 m/s²). The tension in the rope, \( T \), must balance this force to maintain constant velocity.
Set up the equation for equilibrium: Since the net force is zero, \( T - F_g = 0 \). Rearrange this to find \( T = F_g \).
Substitute the values into the equation: Replace \( m \) with 50 kg and \( g \) with 9.8 m/s² in the equation \( T = m \cdot g \) to calculate the tension in the rope.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 the weight of the box and provide the necessary force to maintain its steady upward motion.
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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. In this case, since the box moves at a constant velocity (steady 5.0 m/s), the net force is zero, indicating that the upward tension equals the downward gravitational force.
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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 (approximately 9.81 m/s² on Earth). For the 50 kg box, the weight is 50 kg × 9.81 m/s², which is essential for determining the tension in the rope when the box is in motion.
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Related Practice
Textbook Question
FIGURE EX6.10 shows the force acting on a 2.0 kg object as it moves along the x-axis. The object is at rest at the origin at t = 0 s. What are its acceleration and velocity at t = 6 s?
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Textbook Question
A horizontal rope is tied to a 50 kg box on frictionless ice. What is the tension in the rope if: The box moves at a steady 5.0 m/s?
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Textbook Question
A 50 kg box hangs from a rope. What is the tension in the rope if: The box has vy = 5.0 m/s and is speeding up at 5.0 m/s2
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Textbook Question
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?
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Textbook Question
A woman has a mass of 55 kg. What is her weight while standing on earth?
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