Problem 58 - College Physics

Question

IE . co Two blocks are connected by a light string and accelerated upward by a pulling force \(F\). The mass of the upper block is \(50.0 \mathrm{~kg}\) and that of the lower block is \(100 \mathrm{~kg}\). The upward acceleration of the system as a whole is \(1.50 \mathrm{~m} / \mathrm{s}^{2} .\) Neglect the mass of the string. (a) Draw the free-body diagram of each block. Use the diagrams to determine which of the following is true for the magnitude of the string tension \(T\) compared to other forces: \((1) T>w_{2}\) and \(Tw_{2}\) and \(T>F ;\) (3) \(T

Step-by-Step Solution

Answer

(a) Option 1: \(T>w_2\) and \(T

Step 1: Draw Free-Body Diagrams

For the upper block with mass 50.0 kg, the forces are: - Weight of the block, \(w_1 = 50.0\, \mathrm{kg} \times 9.8\, \mathrm{m/s}^2 = 490\, \mathrm{N}\) downward. - Tension in the string, \(T\) downward. - Pulling force, \(F\) upward. For the lower block with mass 100 kg, the forces are: - Weight of the block, \(w_2 = 100\, \mathrm{kg} \times 9.8\, \mathrm{m/s}^2 = 980\, \mathrm{N}\) downward.- Tension in the string, \(T\) upward.Since the system accelerates upward, for the lower block, the upward acceleration and the tension suggest that \(T\) must be greater than \(w_2\), and for the upper block, \(T\) must be less than \(F\). Hence, option 1 is correct: \(T>w_2\) and \(T

Step 2: Apply Newton's Second Law to the System

To find the required pulling force \(F\), consider the entire system of both blocks as a single object with a combined mass of \(50.0\, \mathrm{kg} + 100\, \mathrm{kg} = 150\, \mathrm{kg}\). The total weight of the system is: \(w = 150\, \mathrm{kg} \times 9.8\, \mathrm{m/s}^2 = 1470\, \mathrm{N}\).Applying Newton's second law for the system:\[ F - w = ma \]\[ F - 1470\, \mathrm{N} = 150\, \mathrm{kg} \times 1.50\, \mathrm{m/s}^2 \]\[ F - 1470\, \mathrm{N} = 225\, \mathrm{N} \]\[ F = 1695\, \mathrm{N} \]So, the required pulling force \(F = 1695\, \mathrm{N}\).

Step 3: Calculate Tension in the String

Apply Newton’s second law to the lower block:For the lower block:\[ T - w_2 = ma_2 \]\[ T - 980\, \mathrm{N} = 100\, \mathrm{kg} \times 1.50\, \mathrm{m/s}^2 \]\[ T - 980\, \mathrm{N} = 150\, \mathrm{N} \]\[ T = 1130\, \mathrm{N} \]The tension in the string, \(T = 1130\, \mathrm{N}\).

Key Concepts

Free-Body Diagrams: Analyzing Forces on Each Block

In physics, a free-body diagram is a simple graphical illustration used to visualize the forces acting on a single object. Consider it as a tool that helps break down complex problems into simpler parts.
To create a free-body diagram, follow these steps:

• Identify the object you want to analyze: In our case, two blocks connected by a string being pulled.
• Draw the object as a simple shape: a box often works well.
• Represent all forces acting on the object with arrows:
  • The length and direction of an arrow show the force's magnitude and direction.
  • Label each force explicitly to avoid confusion.

For the upper block (50 kg), we have three main forces:

• Weight (\(w_1 = 490 \, \mathrm{N}\)) acting downward.
• Tension (\(T\)) in the string acting downward.
• Pulling force (\(F\)) acting upward.

For the lower block (100 kg):

• Weight (\(w_2 = 980 \, \mathrm{N}\)) acting downward.
• Tension (\(T\)) acting upward.

Using these diagrams, we deduce that for the system to move upward, the pulling force (\(F\)) must be greater than the total weight, and the tension (\(T\)) must be such that it is less than the pulling force but greater than the weight of the lower block.

The Role of Tension in the System

Tension is the force exerted by a string, rope, or wire when it is pulled tight. It plays a crucial role in systems where objects are connected, as seen in this exercise with the two blocks.
Tension must overcome gravitational forces acting on the masses while aiding the pulling force. Its magnitude can be critical for determining whether the system will move in the desired direction.

• For the lower block, tension (\(T\)) acts upward, counteracting the block's weight.
• For the upper block, tension acts downward beside its weight, thus requiring the pulling force to be greater for upward acceleration.

By correctly applying Newton's laws and equilibrium conditions on the free-body diagrams, we can solve for tension. This conceptual understanding helps in systematically solving for tension when given values are different.

Understanding String Acceleration

In this exercise, the term "string acceleration" refers to the acceleration of the connected blocks, imparted via the string. When a force pulls on the string, it transmits this force through tension, causing the blocks to accelerate.
This acceleration reflects how the motion is distributed in a connected system. Here, both blocks accelerate equally because they're connected by an ideal string (massless and non-stretching).

• The upward acceleration of the system is given as \(1.50 \, \mathrm{m/s^2}\).
• All calculations regarding forces on the blocks use this common acceleration, as it's the same for every object in the system.

The string serves as a medium for this acceleration, allowing the force to be tunneled effectively to both the lower and upper block, thereby keeping the system's dynamics stable and predictable.

Applying Newton's Second Law: Calculating Forces

Newton's second law establishes a fundamental relationship between an object's mass (\(m\)), its acceleration (\(a\)), and the net force (\(F\)) acting upon it—given by the equation: \(F = ma\).
When analyzing systems like the blocks in this exercise, Newton's second law becomes an indispensable tool.

• For the whole system (150 kg), the total net force includes both the pulling force (\(F\)) and the gravitational force against the system.
• By rearranging the formula, we can solve for unknown quantities such as the pulling force:
  • \(F - w = ma\), where \(w\) is the gravitational force (system's weight).

This approach facilitates determining the required pulling force and understanding how forces have to align to achieve the desired motion. The same principle applies to determine the tension in the string affecting each block distinctly.

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Problem 59

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Key Concepts

Free-Body Diagrams: Analyzing Forces on Each Block

In physics, a free-body diagram is a simple graphical illustration used to visualize the forces acting on a single object. Consider it as a tool that helps break down complex problems into simpler parts.
To create a free-body diagram, follow these steps:

• Identify the object you want to analyze: In our case, two blocks connected by a string being pulled.
• Draw the object as a simple shape: a box often works well.
• Represent all forces acting on the object with arrows:
  • The length and direction of an arrow show the force's magnitude and direction.
  • Label each force explicitly to avoid confusion.

For the upper block (50 kg), we have three main forces:

• Weight (\(w_1 = 490 \, \mathrm{N}\)) acting downward.
• Tension (\(T\)) in the string acting downward.
• Pulling force (\(F\)) acting upward.

For the lower block (100 kg):

• Weight (\(w_2 = 980 \, \mathrm{N}\)) acting downward.
• Tension (\(T\)) acting upward.

Using these diagrams, we deduce that for the system to move upward, the pulling force (\(F\)) must be greater than the total weight, and the tension (\(T\)) must be such that it is less than the pulling force but greater than the weight of the lower block.

The Role of Tension in the System

Tension is the force exerted by a string, rope, or wire when it is pulled tight. It plays a crucial role in systems where objects are connected, as seen in this exercise with the two blocks.
Tension must overcome gravitational forces acting on the masses while aiding the pulling force. Its magnitude can be critical for determining whether the system will move in the desired direction.

• For the lower block, tension (\(T\)) acts upward, counteracting the block's weight.
• For the upper block, tension acts downward beside its weight, thus requiring the pulling force to be greater for upward acceleration.

By correctly applying Newton's laws and equilibrium conditions on the free-body diagrams, we can solve for tension. This conceptual understanding helps in systematically solving for tension when given values are different.

Understanding String Acceleration

In this exercise, the term "string acceleration" refers to the acceleration of the connected blocks, imparted via the string. When a force pulls on the string, it transmits this force through tension, causing the blocks to accelerate.
This acceleration reflects how the motion is distributed in a connected system. Here, both blocks accelerate equally because they're connected by an ideal string (massless and non-stretching).

• The upward acceleration of the system is given as \(1.50 \, \mathrm{m/s^2}\).
• All calculations regarding forces on the blocks use this common acceleration, as it's the same for every object in the system.

The string serves as a medium for this acceleration, allowing the force to be tunneled effectively to both the lower and upper block, thereby keeping the system's dynamics stable and predictable.

Applying Newton's Second Law: Calculating Forces

Newton's second law establishes a fundamental relationship between an object's mass (\(m\)), its acceleration (\(a\)), and the net force (\(F\)) acting upon it—given by the equation: \(F = ma\).
When analyzing systems like the blocks in this exercise, Newton's second law becomes an indispensable tool.

• For the whole system (150 kg), the total net force includes both the pulling force (\(F\)) and the gravitational force against the system.
• By rearranging the formula, we can solve for unknown quantities such as the pulling force:
  • \(F - w = ma\), where \(w\) is the gravitational force (system's weight).

This approach facilitates determining the required pulling force and understanding how forces have to align to achieve the desired motion. The same principle applies to determine the tension in the string affecting each block distinctly.