Question

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

Answer 1

Let `M` be mass of block and `m` be mass suspended with an inextensible string, `mu` be coefficient of friction, `theta` be angle made by string with the horizontal where `theta>=0` and `T` be tension, (reaction force) in the strings. It is given that block has a movement. Let `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`-axis and North as positive `y`-axis.

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

`(M+m)a=mgcostheta-mu(Mg-mgsintheta)` ......(1)

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

`a=T_x/M`
`=>a=(Tcostheta)/M`
`=>T=(Ma)/costheta`
`=>T=(M(mgcostheta-mu(Mg-mgsintheta)))/((M+m)costheta)` .....(2)

Rewriting it as

`T=a-b/costheta+ctantheta`
where `a,b and c` are system parameters defined with help of (2) not dependent on `theta`

We see that `T` is dependent on two terms involving `theta`

1. `-1/costheta`. For `T` to be a smaller number `costheta` term must be maximum. We know that `costheta` has a maximum value `=1` for `theta=0^@`
2. `tantheta`. For `T` to be a smaller number, `tantheta` term must be zero. We know that `tantheta` has a value `=0` for `theta=0^@`.

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