Question

A tool chest has 650 N weight that acts through the midpoint of the chest. The chest is supported by feet at A and rollers at B. The surface has a coefficient of friction of 0.3. Determine the value of the horizontal force P necessary to cause motion of the chest to the right, and determine if the motion is sliding or tipping.

Answer 1

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.

Step-by-step 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

`F_f= \mu N_2`

where :

`F_f =` friction force

`\mu` = coefficient of friction

`N_2` = normal friction

Given that:

`\mu` = 0.3

`F_f =` 0.3
`N_2`

Using the equation of equilibrium along horizontal direction.

`\sum f_x = 0`

P -
`F_f =` 0

P = 0.3
`N_2` ----- Equation (1)

To determine the moment about point B ; we have the expression

`\sum M_B = 0`

0 =
`N_2*70-W*35-P*100`

where;

P = horizontal force

`N_2` = normal force at support A

W = self- weight of tool chest

Replacing W = 650 N

0 =
`N_2*70-650*35-100*P`

`P = (70 N_2-22750)/(100) ----- equation (2)`

Replacing
`(70 N_2-22750)/(100)` for P in equation (1)

`(70N_2 -22750)/(100) =0.3 N_2`

`N_2 = (22750)/(40)`
`N_2 = 568.75 \ N`

Plugging the value of
`N_2 = 568.75 \ N` in equation (2)

`P = (70(568.75)-22750)/(100) \\ \\ P = (39812.5-22750)/(100) \\ \\ P = (17062.5)/(100)`

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:

`\sum M_A = 0`

-(P × 100)+ (W×35) = 0

P =
`(W*35)/(100)`

Replacing 650 N for W

`P = (650*35)/(100)`

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.