Why would tension be smaller if the string were parallel to the lab bench?

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Why would tension be smaller if the string were parallel to the lab bench?

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Answer 1 · 2018-05-30T11:52:43

Let $M$ $M$ be mass of block and $m$ $m$ be mass suspended with an inextensible string, $μ$ $mu$ be coefficient of friction, $θ$ $theta$ be angle made by string with the horizontal where $θ \geq 0$ $theta>=0$ and $T$ $T$ be tension, (reaction force) in the strings. It is given that block has a movement. Let $a$ $a$ be its acceleration. As both masses are joined with a common string, the hanging mass also moves downwards with the same acceleration.
Taking East as positive $x$ $x$ -axis and North as positive $y$ $y$ -axis.

External forces responsible for the magnitude of acceleration of masses when considered as single object

$(M + m) a = m g cos θ - μ (M g - m g sin θ)$ $(M+m)a=mgcostheta-mu(Mg-mgsintheta)$ ......(1)

For Block it is $x$ $x$ component of tension which is responsible for its acceleration.

$a = \frac{T_{x}}{M}$ $a=T_x/M$
$\Rightarrow a = \frac{T cos θ}{M}$ $=>a=(Tcostheta)/M$
$\Rightarrow T = \frac{M a}{cos θ}$ $=>T=(Ma)/costheta$
$\Rightarrow T = \frac{M (m g cos θ - μ (M g - m g sin θ))}{(M + m) cos θ}$ $=>T=(M(mgcostheta-mu(Mg-mgsintheta)))/((M+m)costheta)$ .....(2)

Rewriting it as

$T = a - \frac{b}{cos θ} + c tan θ$ $T=a-b/costheta+ctantheta$
where $a, b and c$ $a,b and c$ are system parameters defined with help of (2) not dependent on $θ$ $theta$

We see that $T$ $T$ is dependent on two terms involving $θ$ $theta$

$- \frac{1}{cos θ}$ $-1/costheta$ . For $T$ $T$ to be a smaller number $cos θ$ $costheta$ term must be maximum. We know that $cos θ$ $costheta$ has a maximum value $= 1$ $=1$ for $θ = 0^{\circ}$ $theta=0^@$
$tan θ$ $tantheta$ . For $T$ $T$ to be a smaller number, $tan θ$ $tantheta$ term must be zero. We know that $tan θ$ $tantheta$ has a value $= 0$ $=0$ for $θ = 0^{\circ}$ $theta=0^@$ .

Hence, we see that tension will be smaller if the string connecting the block were parallel to the lab bench.

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Why would tension be smaller if the string were parallel to the lab bench?

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