HW5 Prob2

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Vladimir Vingoradov

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7 years, 4 months ago

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Vladimir Vingoradov 4 years, 4 months ago

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Vladimir Vingoradov 7 years, 4 months ago

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Vladimir Vingoradov 7 years, 5 months ago

Created this as a copy of HW5 Prob2.

Name	Status	Author	Last Modified
HW5 Prob1	Ready to use	Vladimir Vingoradov	11/03/2025 17:07
HW5 Prob2	Ready to use	Vladimir Vingoradov	02/12/2024 14:55
Friction: Range of values 1	Ready to use	William Haynes	10/04/2023 10:42
Friction: Determine direction of $F_f$	Ready to use	William Haynes	21/05/2023 16:56

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Question statement

Knowing that $\theta=\var{th}^\circ$ , determine the range of values of $P$ for which equilibrium is maintained.

Give any introductory information the student needs.

			Name	Type	Generated Value

th	integer	55
mu_s	number	0.2
mu_k	number	0.15
Pmin	number	252.1
Pmax	number	500.1
phi	number	11.31

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random(10..70#5 except 25)

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Generated value: `integer`

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Pmax
Pmin

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$+\!\!\nearrow\sum F_y=0,\quad\implies\quad N-800 \cos 25^\circ+P\sin(\theta-25^\circ)=0$

$N=800 \cos 25^\circ-P\sin( \var{th-25}^\circ),$

$F_m=\mu_s N$

To start moving the block up the incline:

$+\!\!\searrow\sum F_x=0,\quad\implies\quad \mu_s N-P\cos(\theta-25^\circ)+800\sin 25^\circ=0$

$\mu_s (800 \cos 25^\circ-P\sin\var{th-25}^\circ)-P\cos\var{th-25}^\circ+800\sin 25^\circ=0$

$\mu_s 800 \cos 25^\circ-\mu_s P\sin\var{th-25}^\circ-P\cos\var{th-25}^\circ+800\sin 25^\circ=0$

$\mu_s 800 \cos 25^\circ+800\sin 25^\circ=+P[\mu_s \sin\var{th-25}^\circ+\cos\var{th-25}^\circ]$

$P=\dfrac{800(\sin 25^\circ+\mu_s\cos 25^\circ)}{\cos\var{th-25}^\circ+\mu_s \sin\var{th-25}^\circ}=\var{Pmax}\,\text{N}\quad\blacktriangleleft$

To prevent the block from moving down:

$+\!\!\searrow\sum F_x=0,\quad\implies\quad -\mu_s N-P\cos(\theta-25^\circ)+800\sin 25^\circ=0$

$-\mu_s (800 \cos 25^\circ-P\sin \var{th-25}^\circ)-P\cos\var{th-25}^\circ+800\sin 25^\circ=0$

$-\mu_s 800 \cos 25^\circ +\mu_s P\sin \var{th-25}^\circ- P\cos \var{th-25}^\circ+800\sin 25^\circ=0$

$\mu_s P\sin \var{th-25}^\circ- P\cos\var{th-25}^\circ-\mu_s 800 \cos 25^\circ +800\sin 25^\circ=0$

$-\mu_s 800 \cos 25^\circ +800\sin 25^\circ=P[\cos\var{th-25}^\circ-\mu_s \sin\var{th-25}^\circ]$

$P=\dfrac{800(\sin 25^\circ-\mu_s\cos 25^\circ)}{\cos\var{th-25}^\circ-\mu_s \sin\var{th-25}^\circ}=\var{Pmin}\,\text{N}\quad\blacktriangleleft$

$\var{Pmin}\,\text{N} \le P \le \var{Pmax}\,\text{N}\quad\blacktriangleleft$

Alternative solution:

Combine the normal and friction forces $\vec{R}=\vec{N}+\vec{F}_\mu$ . When $F_\mu=\mu_sN$ , the angle between the resultant $\vec{R}$ and the normal force is the friction angle:

$\phi_s=\arctan \mu_s=\arctan \var{mu_s}=\var{phi}^\circ.$

To prevent the block from moving up:

In this case there are only three forces acting on the block, which in view of equilibrium, form a triangle:

In this triangle, the angle between the force $\vec{R}$ and weight $\vec{W}$ is $\phi_s+25^\circ$ . Using the law of sines:

$\displaystyle\frac{P}{\sin (\phi_s+25^\circ)}=\frac{W}{\sin \left[180^\circ-(\phi_s+25^\circ)-(90^\circ-\theta)\right]},$

$\displaystyle\frac{P}{\sin\var{phi+25}^\circ}=\frac{800}{\sin\var{180-phi-25-90+th}^\circ},$

$P=800\dfrac{\sin\var{phi+25}^\circ}{\sin\var{180-phi-25-90+th}^\circ}=\var{Pmax}\,\text{N}.\,\blacktriangleleft$

To prevent the block from moving down send the friction force up the incline:

Form a triangle:

In this triangle, the angle between the force $\vec{R}$ and weight $\vec{W}$ is $25^\circ-\phi_s$ . Using the law of sines:

$\displaystyle\frac{P}{\sin (25^\circ-\phi_s)}=\frac{W}{\sin \left[180^\circ-(25^\circ-\phi_s)-(90^\circ-\theta)\right]},$

$\displaystyle\frac{P}{\sin\var{-phi+25}^\circ}=\frac{800}{\sin\var{180+phi-25-90+th}^\circ},$

$P=800\dfrac{\sin\var{-phi+25}^\circ}{\sin\var{180+phi-25-90+th}^\circ}=\var{Pmin}\,\text{N}\,\blacktriangleleft$

$\var{Pmin}\,\text{N} \le P \le \var{Pmax}\,\text{N}.\quad\blacktriangleleft$

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This question is used in the following exams:

Homework 5 Friction + Solutions by Vladimir Vingoradov in SFY0010.
tutorial: fiction by Vladimir Vingoradov in ENG1507 Mechanics 1.

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Vladimir Vingoradov 4 years, 4 months ago

Vladimir Vingoradov 7 years, 4 months ago

Vladimir Vingoradov 7 years, 5 months ago

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HW5 Prob2

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Comment

Checkpoint description

Vladimir Vingoradov 4 years, 4 months ago

Vladimir Vingoradov 7 years, 4 months ago

Vladimir Vingoradov 7 years, 5 months ago

Error in variable testing condition

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Generated value: integer

Parts

Number entry

Advanced settings

Generated value: `integer`