Question

The spring is unstretched at the position x = 0. under the action of a force p, the cart moves from the initial position x1 = -8 in. to the final position x2 = 5 in. determine (a) the work done on the cart by the spring and (b) the work done on the cart by its weight.

Answer 1

This question is incomplete, the missing diagram is uploaded along this Answer below.

Answer:

a) the work done on the cart by the spring is 4.875 lb-ft

b) the work done on the cart by its weight is - 3.935 lb-ft

Explanation:

Given the data in the question;

(a) determine the work done on the cart by the spring

we calculate the work done on the cart by the spring as follows;

$W_{spring}$ = 1/2×k( $x^{2} _{1}$ - $x^{2} _{2}$ )

where k is spring constant ( 3 lb/in )

we substitute  

$W_{spring}$ = 1/2 × 3( (-8)² - (5)² )      

$W_{spring}$ = 1/2 × 3( 64 - 25 )

$W_{spring}$ = 1/2 × 3( 39 )

$W_{spring}$ = 58.5 lb-in

we convert to pound force-foot

$W_{spring}$ = 58.5 × 0.0833333 lb-ft

$W_{spring}$ = 4.875 lb-ft

Therefore, the work done on the cart by the spring is 4.875 lb-ft

b) the work done on the cart by its weight

work done by its weight;

$W_{gravity}$ = -mgsin∅( x₂ - x₁ )        

we substitute in of values from the image below;

$W_{gravity}$ = -14 × sin(15°)( 5 - (-8) )  

$W_{gravity}$ = -14 × 0.2588 × 13

$W_{gravity}$ = -47.1  lb-in

we convert to pound force-foot

$W_{gravity}$ = -47.1 × 0.0833333 lb-ft

$W_{gravity}$ = - 3.935 lb-ft

Therefore, the work done on the cart by its weight is - 3.935 lb-ft