A rectangular box lies on a rough inclined surface. The co-efficient of friction between the surface and the box is µ. Let the mass of the box be m.

(a) At what angle of inclination θ of the plane to the horizontal will the box just start to slide down the plane?

(b) What is the force acting on the box down the plane, if the angle of inclination of the plane is increased to α θ > ?

(c) What is the force needed to be applied upwards along the plane to make the box either remain stationary or just move up with uniform speed?

(d) What is the force needed to be applied upwards along the plane to make the box move up the plane with acceleration a?

This is a Long Answer type Questions as classified in NCERT Exemplar

Explanation -for the box to just starts sliding down mg

sin $\theta =t=\mu N=\mu mgcos\theta$

tan $\theta$ = $\mu$

$\theta ={tan}^{-1}\left(\mu \right)$

b) when angle of inclination is increased to $\alpha >\theta$

F₁= mgsin $\alpha -f=mgsin\alpha -\mu N$

= mg ( $sin\alpha -\mu cos\alpha$ )

c) to keep the box stationary, upward force needed F₂= m $gsin\alpha$ +f= mg ( $sin\alpha +\mu cos\alpha$ )

d) if the box is to move with an upward acceleration a then upward force needed 

F₃=mg ( $sin\alpha +\mu cos\alpha$ )+ma

 

<p><span lang="EN-IN">This is a Long Answer type Questions as classified in NCERT Exemplar</span></p><p><strong><span lang="EN-IN">Explanation -</span></strong><span lang="EN-IN">for the box to just starts sliding down mg</span></p><p><span lang="EN-IN">sin<span class="mathml" contenteditable="false"> <math> <mi>θ</mi> <mo>=</mo> <mi>t</mi> <mo>=</mo> <mi>μ</mi> <mi>N</mi> <mo>=</mo> <mi>μ</mi> <mi>m</mi> <mi>g</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>θ</mi> </math> </span></span><span lang="EN-IN">&lt;!- [endif]-&gt;</span>&lt;!- [endif]-&gt;</p><div class="photo-widget-full"><div class="figure"><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1751263301phpwFHuPD.jpeg" alt="" width="339" height="180" loading="lazy"></picture></div><div class="figure"><p><span lang="EN-IN">tan<span class="mathml" contenteditable="false"> <math> <mi>θ</mi> </math> </span></span><span lang="EN-IN">=<span class="mathml" contenteditable="false"> <math> <mi>μ</mi> </math> </span></span><span lang="EN-IN">&lt;!- [endif]-&gt;</span>&lt;!- [endif]-&gt;</p><p><span lang="EN-IN"><span class="mathml" contenteditable="false"> <math> <mi>θ</mi> <mo>=</mo> <msup> <mrow> <mrow> <mi>t</mi> <mi>a</mi> <mi>n</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </msup> <mo> (</mo> <mi>μ</mi> <mo>)</mo> </math> </span><br>&lt;!- [endif]-&gt;</span>&lt;!- [endif]-&gt;</p><p><span lang="EN-IN">b) when angle of inclination is increased to <span class="mathml" contenteditable="false"> <math> <mi>α</mi> <mo>&gt;</mo> <mi>θ</mi> </math> </span></span><span lang="EN-IN">&lt;!- [endif]-&gt;</span>&lt;!- [endif]-&gt;</p><p><span lang="EN-IN">F₁= mgsin <span class="mathml" contenteditable="false"> <math> <mi>α</mi> <mo>-</mo> <mi>f</mi> <mo>=</mo> <mi>m</mi> <mi>g</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>α</mi> <mo>-</mo> <mi>μ</mi> <mi>N</mi> </math> </span></span><span lang="EN-IN">&lt;!- [endif]-&gt;</span>&lt;!- [endif]-&gt;</p><p><span lang="EN-IN">= mg (<span class="mathml" contenteditable="false"> <math> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>α</mi> <mo>-</mo> <mi>μ</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>α</mi> </math> </span></span><span lang="EN-IN">)</span></p><p><span lang="EN-IN">c) to keep the box stationary, upward force needed F₂= m<span class="mathml" contenteditable="false"> <math> <mi>g</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>α</mi> </math> </span></span><span lang="EN-IN">+f= mg (<span class="mathml" contenteditable="false"> <math> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>α</mi> <mo>+</mo> <mi>μ</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>α</mi> </math> </span></span><span lang="EN-IN">)</span></p><p><span lang="EN-IN">d) if the box is to move with an upward acceleration a then upward force needed&nbsp;</span></p><p><span lang="EN-IN">F₃=mg (<span class="mathml" contenteditable="false"> <math> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>α</mi> <mo>+</mo> <mi>μ</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>α</mi> </math> </span></span><span lang="EN-IN">)+ma</span></p></div></div><p>&nbsp;</p>

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