Answer:
the value of horizontal force P is 170.625 N
the value of horizontal force at P = 227.5 N is that the block moves to right and this motion is due to sliding.
Explanation:
The first diagram attached below shows the free body diagram of the tool chest when it is sliding.
Let start out by calculating the friction force
[tex]F_f= \mu N_2[/tex]
where :
[tex]F_f =[/tex] friction force
[tex]\mu[/tex] = coefficient of friction
[tex]N_2[/tex] = normal friction
Given that:
[tex]\mu[/tex] = 0.3
[tex]F_f =[/tex] 0.3 [tex]N_2[/tex]
Using the equation of equilibrium along horizontal direction.
[tex]\sum f_x = 0[/tex]
P - [tex]F_f =[/tex] 0
P = 0.3 [tex]N_2[/tex] ----- Equation (1)
To determine the moment about point B ; we have the expression
[tex]\sum M_B = 0[/tex]
0 = [tex]N_2*70-W*35-P*100[/tex]
where;
P = horizontal force
[tex]N_2[/tex] = normal force at support A
W = self- weight of tool chest
Replacing W = 650 N
0 = [tex]N_2*70-650*35-100*P[/tex]
[tex]P = \frac{70 N_2-22750}{100} ----- equation (2)[/tex]
Replacing [tex]\frac{70 N_2-22750}{100}[/tex] for P in equation (1)
[tex]\frac{70N_2 -22750}{100} =0.3 N_2[/tex]
[tex]N_2 = \frac{22750}{40}[/tex][tex]N_2 = 568.75 \ N[/tex]
Plugging the value of [tex]N_2 = 568.75 \ N[/tex] in equation (2)
[tex]P = \frac{70(568.75)-22750}{100} \\ \\ P = \frac{39812.5-22750}{100} \\ \\ P = \frac{17062.5}{100}[/tex]
P =170.625 N
Thus; the value of horizontal force P is 170.625 N
b) From the second diagram attached the free body diagram; the free body diagram of the tool chest when it is tipping about point A is also shown below:
Taking the moments about point A:
[tex]\sum M_A = 0[/tex]
-(P × 100)+ (W×35) = 0
P = [tex]\frac{W*35}{100}[/tex]
Replacing 650 N for W
[tex]P = \frac{650*35}{100}[/tex]
P = 227.5 N
Thus; the value of horizontal force P, when the tool chest tipping about point A is 227.5 N
We conclude that the motion will be impending for the lowest value when P = 170.625 N and when P= 227.5 N
However; the value of horizontal force at P = 227.5 N is that the block moves to right and this motion is due to sliding.
