Lexi Holmes

2023-04-01

Three identical blocks connected by ideal strings are being pulled along a horizontal frictionless surface by a horizontal force $\stackrel{\to }{F}$ The magnitude of the tension in the string between blocks B and C is T=3.00N. Each block has mass m=0.400kg.
What is the magnitude F of the force?
What is the tension in the string between block A and block B??

Santino Ray

To solve this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration.
Let's denote the tension in the string between blocks A and B as ${T}_{AB}$, and the tension in the string between blocks B and C as ${T}_{BC}$. The force applied is denoted as $F$, and the mass of each block is denoted as $m$.
1. To find the magnitude of the force $F$, we need to consider the motion of the entire system. Since the blocks are connected by ideal strings, they will all have the same acceleration. Therefore, we can write the equation of motion for any one of the blocks. Let's choose block A.
The net force acting on block A is given by:
$F-{T}_{AB}=m·a$
Since the tension in the string between block A and block B is unknown, we can rewrite this equation as:
$F-{T}_{AB}=m·{a}_{1}$
where ${a}_{1}$ represents the acceleration of block A.
2. To find the tension in the string between block A and block B, we can consider the motion of block B. The net force acting on block B is given by:
${T}_{AB}-{T}_{BC}=m·{a}_{2}$
where ${a}_{2}$ represents the acceleration of block B.
Since the blocks are identical and connected by ideal strings, their accelerations ${a}_{1}$ and ${a}_{2}$ are the same. Therefore, we can rewrite the above equation as:
${T}_{AB}-{T}_{BC}=m·{a}_{1}$
Now, we can substitute the given values into the equations. We have ${T}_{AB}=3.00\phantom{\rule{0.167em}{0ex}}\text{N}$ and $m=0.400\phantom{\rule{0.167em}{0ex}}\text{kg}$.
From equation (2), we can solve for ${T}_{BC}$:
${T}_{BC}={T}_{AB}-m·{a}_{1}$
${T}_{BC}=3.00\phantom{\rule{0.167em}{0ex}}\text{N}-\left(0.400\phantom{\rule{0.167em}{0ex}}\text{kg}\right)·{a}_{1}$
Substituting this expression for ${T}_{BC}$ into equation (1), we can solve for the force $F$:
$F-{T}_{AB}=m·{a}_{1}$
$F-3.00\phantom{\rule{0.167em}{0ex}}\text{N}=\left(0.400\phantom{\rule{0.167em}{0ex}}\text{kg}\right)·{a}_{1}$
Since the blocks are being pulled along a horizontal frictionless surface, the acceleration ${a}_{1}$ is equal to the magnitude of the force $F$ divided by the total mass of the system:
${a}_{1}=\frac{F}{3·m}$
Substituting this into the equation above, we can solve for $F$:
$F-3.00\phantom{\rule{0.167em}{0ex}}\text{N}=\left(0.400\phantom{\rule{0.167em}{0ex}}\text{kg}\right)·\left(\frac{F}{3·m}\right)$
Simplifying the equation, we get:
$F-3.00\phantom{\rule{0.167em}{0ex}}\text{N}=\frac{F}{3}$
Solving for $F$, we find:
$\frac{2F}{3}=3.00\phantom{\rule{0.167em}{0ex}}\text{N}$
$F=\frac{3}{2}·3.00\phantom{\rule{0.167em}{0ex}}\text{N}$
$F=
4.50\phantom{\rule{0.167em}{0ex}}\text{N}$

Therefore, the magnitude of the force $F$ is 4.50 N.
To find the tension in the string between block A and block B, we can use equation (2) and substitute the values we know:
${T}_{AB}-{T}_{BC}=m·{a}_{1}$
$3.00\phantom{\rule{0.167em}{0ex}}\text{N}-{T}_{BC}=\left(0.400\phantom{\rule{0.167em}{0ex}}\text{kg}\right)·\left(\frac{4.50\phantom{\rule{0.167em}{0ex}}\text{N}}{3·\left(0.400\phantom{\rule{0.167em}{0ex}}\text{kg}\right)}\right)$
Simplifying the equation, we get:
$3.00\phantom{\rule{0.167em}{0ex}}\text{N}-{T}_{BC}=1.50\phantom{\rule{0.167em}{0ex}}\text{N}$
Solving for ${T}_{BC}$, we find:
${T}_{BC}=3.00\phantom{\rule{0.167em}{0ex}}\text{N}-1.50\phantom{\rule{0.167em}{0ex}}\text{N}$
${T}_{BC}=1.50\phantom{\rule{0.167em}{0ex}}\text{N}$
Therefore, the tension in the string between block A and block B is 1.50 N.

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