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Chapter 5: Problem 4

A block of mass \(0 \cdot 2 \mathrm{~kg}\) is suspended from the ceiling by a light string. A second block of mass \(0 \cdot 3 \mathrm{~kg}\) is suspended from the first block through another string. Find the tensions in the two strings. Take \(g=10 \mathrm{~m} / \mathrm{s}^{2}\).

Short Answer

Expert verified
Tension at the top string (T1) is 5 N, and tension at the second string (T2) is 3 N.

Step by step solution

01

Identify the Forces

First, we need to identify the forces acting on each block. The top block (0.2 kg) experiences tension from the string holding it (T1) and the gravitational force acting downward, which is its weight (W1 = mg = 0.2kg * 10m/s虏 = 2 N). The bottom block (0.3 kg) has tension from the string between the blocks pulling upward (T2) and its weight acting downward (W2 = mg = 0.3kg * 10m/s虏 = 3 N).
02

Analyze the Forces on the Bottom Block

For the lower block, we have the gravitational force acting downward and tension (T2) acting upward. The sum of the forces acting on it equals zero since the system is in equilibrium. Hence, we have: \[ T2 - W2 = 0 \]Substitute W2 = 3 N to get: \[ T2 = 3 \text{ N} \]
03

Analyze the Forces on the Top Block

For the upper block, tension T1 holds up both the top block and the lower string's tension. Thus, the forces acting on the top block are the gravitational force of the block itself and the tension from the string below it. Therefore:\[ T1 = W1 + T2 \]Substitute W1 = 2 N and T2 = 3 N to get:\[ T1 = 2 \text{ N} + 3 \text{ N} = 5 \text{ N} \]
04

Verify the Solution

We have calculated the forces and assumed equilibrium as the blocks are not accelerating. Check by summing upward and downward forces to ensure all equations are satisfied, confirming T1 and T2 values are consistent with equilibrium.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Free Body Diagrams
Free Body Diagrams are essential for visualizing the forces acting on an object. They help simplify complex problems by displaying all the forces on each block or object in question. It's like drawing a picture of what's happening to an object in a physical situation. This can be incredibly helpful, especially when dealing with blocks and tensions in strings.
  • First, isolate the object you are analyzing. In our example, we are looking at two blocks suspended by strings.
  • Identify all the forces acting on the object. For each block, these forces include gravitational force (weight) and tension.

Once you have your free body diagram, it is easier to write down the equations according to Newton's laws to solve for unknowns, like tension in the strings here. Remember, each force in the diagram is a vector and requires direction and magnitude for complete representation.
Equilibrium of Forces
Understanding the Equilibrium of Forces is key when analyzing systems like a set of suspended blocks. When a system is in equilibrium, the net force acting on it is zero. This means the blocks are not accelerating because the forces pulling them in each direction balance out perfectly.
To analyze such situations:
  • For each block, write down the forces acting on it.
  • The sum of forces in any direction should be zero. This is due to Newton's First Law, which states an object at rest remains at rest unless acted upon by a non-zero net force.

In the step-by-step solution, equilibrium is verified by ensuring the sum of upward forces equals the sum of downward forces for each block. Especially when tension is involved, understanding equilibrium helps you determine the correct values for tension in connected objects.
Tension in Strings
Tension in strings plays a critical role in systems where objects are suspended, like in this exercise. Tension is the force that is transmitted through the string. It pulls equally on the objects at either end of the string, assuming the string is light and unstretchable.
For the block system:
  • The tension in the string attached to the lower block (T2) counteracts the weight of the block, calculated as 3 N.
  • The tension in the string attached to the upper block (T1) must counteract both the weight of the upper block and the tension from the lower string, calculated as 5 N.
Understanding how tension works in these systems and how to use it in calculations allows you to solve many real-life physics problems effectively. Just remember: the tension is always directed along the string, away from the body on which it acts.

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Most popular questions from this chapter

A monkey of mass \(15 \mathrm{~kg}\) is climbing on a rope with one end fixed to the ceiling. If it wishes to go up with an acceleration of \(1 \mathrm{~m} / \mathrm{s}^{2}\), how much force should it apply to the rope? If the rope is \(5 \mathrm{~m}\) long and the monkey starts from rest, how much time will it take to reach the ceiling?

A person is standing on a weighing machine placed on the floor of an elevator. The elevator starts going up with some acceleration, moves with uniform velocity for a while and finally decelerates to stop. The maximum and the minimum weights recorded are \(72 \mathrm{~kg}\) and \(60 \mathrm{~kg}\). Assuming that the magnitudes of the acceleration and the deceleration are the same, find (a) the true weight of the person and (b) the magnitude of the acceleration. Take \(g=9 \cdot 9 \mathrm{~m} / \mathrm{s}^{2}\).

A force \(\vec{F}=\vec{v} \times \vec{A}\) is exerted on a particle in addition to the force of gravity, where \(\vec{v}\) is the velocity of the particle and \(\vec{A}\) is a constant vector in the horizontal direction. With what minimum speed a particle of mass \(m\) be projected so that it continues to move undeflected with a constant velocity?

Two blocks \(A\) and \(B\) of mass \(m_{A}\) and \(m_{B}\) respectively are kept in contact on a frictionless table. The experimenter pushes the block \(A\) from behind so that the blocks accelerate. If the block \(A\) exerts a force \(F\) on the block \(B\), what is the force exerted by the experimenter on \(A\) ?

A car is speeding up on a horizontal road with an acceleration \(a\). Consider the following situations in the car. (i) A ball is suspended from the ceiling through a string and is maintaining a constant angle with the vertical. Find this angle. (ii) A block is kept on a smooth incline and does not slip on the incline. Find the angle of the incline with the horizontal.
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