One end of a horizontal spring with force constant 130.0 N/m is attached to a vertical wall. A 4.00-kg block sitting on the floor is placed against the spring. The coefficient of kinetic friction between the block and the floor is
μk= 0.400. You apply a constant force F, which is directed toward the wall. F has magnitude of 82.0 N and is directed toward the wall. At the instant that the spring is compressed 80.0 cm, what are

(a) the speed of the block, and
(b) the magnitude and direction of the block's acceleration?

Explanation:

spring constant, K = 130 N/m

mass of block, m = 4 kg

coefficient of friction, μk = 0.4

Force, F = 82 N

compression in spring, d = 80 cm = 0.8 m

(a) Let the speed of the block is v.

Use conservation of energy

F x d = 0.5 Kd² + 0.5mv² + μkmgd

2Fd - Kd² - 2 μk mgd = mv²

4 v² = 2 x 82 x 0.8 - 130 x 0.8 x 0.8 - 2 x 0.4 x 4 x 9.8 x 0.8

4v² = 131.2 - 83.2 - 25.088

v² = 5.728

v = 2.4 m/s

(b)

Net force, Fnet = F - friction force - Kd

Fnet = 82 - 0.4 x 4 x 9.8 - 130 x 0.8

Fnet = - 37.68 N

Acceleration of the block, a = Fnet / m

a = - 37.68 / 4

a = - 9.42 m/s²

the direction of acceleration is opposite to the direction of motion

pythagoras' theorem

square root (-5^2+9^2)

square root (25+81)

square root (106)

angle is tan^-1(9/5)

neg x, pos y