Q: Solution

Initially, spring compression ##x = 0.10\ \text{m}##, block at rest. Spring potential energy: ###U_i = \frac{1}{2} k x^2 = \frac{1}{2} \cdot 200 \cdot (0.10)^2 = 100 \cdot 0.01 = 1\ \text{J}### Initial kinetic energy ##K_i = 0##.

Q: Substitution and Final Values

Substitute ##m = 2##, ##M = 8##, ##\theta = 30^\circ##, ##g = 10##.

Question 1

Solution

Accepted Answer

Initially, spring compression ##x = 0.10\ ext{m}##, block at rest.
Spring potential energy:
###U_i = \frac{1}{2} k x^2 = \frac{1}{2} \cdot 200 \cdot (0.10)^2 = 100 \cdot 0.01 = 1\ ext{J}###
Initial kinetic energy ##K_i = 0##.

Question 2

Solution

Work done by the force:
      ###W = Fd = 20 	imes 5 = 100\ 	ext{J}###

Initial kinetic energy ##K_i = 0## (starts from rest). By Work–Energy Theorem:
      ###W_{	ext{net}} = \Delta K = K_f &#8211; K_i = K_f###
      So:
      ###K_f = 100 = \frac{1}{2} m v^2 = \frac{1}{2} \cdot 4 \cdot v^2 = 2 v^2###
      Hence:
      ###2 v^2 = 100 \Rightarrow v^2 = 50 \Rightarrow v = \sqrt{50} \approx 7.07\ 	ext{m/s}###

So the final speed is about ##7.1\ 	ext{m/s}##.
PROBLEM 7 – ATWOOD MACHINE (TWO MASSES OVER A PULLEY) (MODERATE–HARD)
Two masses, ##m_1 = 4\ 	ext{kg}## and ##m_2 = 6\ 	ext{kg}##, are connected by a light string over a frictionless
    pulley (Atwood machine).

Find the acceleration of the system.

Find the tension in the string.

Take ##g = 10\ 	ext{m/s}^2##.
Laws involved: Newton’s Second Law for each mass, Newton’s Third Law.
Solution

Accepted Answer

Heavier mass ##m_2## goes down, lighter mass ##m_1## goes up.

Question 3

Key Ideas (Short Version)

Accepted Answer

The block slides down the incline with acceleration relative to the wedge ##a_{ ext{rel}}## along the plane. The wedge itself accelerates horizontally because the block exerts a reaction on it. No external horizontal forces act, so the horizontal momentum of the block–wedge system is conserved.

Question 4

Substitution and Final Values

Accepted Answer

Substitute ##m = 2##, ##M = 8##, ##	heta = 30^\circ##, ##g = 10##.

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