To determine the force required to move the metal box on a metal surface under the given conditions, we need to consider two cases:
Case I: Just Moving the Box (Static Friction)
When you’re trying to move the box without causing it to accelerate (just overcoming static friction), you need to apply a force equal to the maximum static friction force. The maximum static friction force is calculated using the formula:
Fstatic=μs⋅NFstatic​=μs​⋅N
Where:

• FstaticFstatic​ is the static friction force.
• μsμs​ is the coefficient of static friction between the box and the surface.
• NN is the normal force, which is equal to the weight of the box (mgmg), where gg is the acceleration due to gravity.

You provided the coefficient of static friction (μsμs​) as 0.5, and the mass of the box (mm) as 4 kg. The acceleration due to gravity (gg) is approximately 9.81 m/s².
First, calculate the normal force (NN):
N=mg=4 kg×9.81 m/s2=39.24 NN=mg=4kg×9.81m/s2=39.24N
Now, calculate the static friction force:
Fstatic=0.5×39.24 N=19.62 NFstatic​=0.5×39.24N=19.62N
So, to just move the box, you need to apply a force of approximately 19.62 N parallel to the surface.
Case II: Moving the Box with an Acceleration of 2 m/s²
In this case, you want to accelerate the box with an acceleration (aa) of 2 m/s². To do this, you need to overcome static friction and provide the additional force required for the acceleration.

1. Overcoming Static Friction: We’ve already calculated the static friction force in Case I, which is 19.62 N. You’ll need to apply this force just to overcome static friction initially.
2. Additional Force for Acceleration: To accelerate the box at 2 m/s², you need to calculate the additional force required using Newton’s second law:
   Facceleration=maFacceleration​=ma
   Where:
   • FaccelerationFacceleration​ is the force required for acceleration.
   • mm is the mass of the box.
   • aa is the desired acceleration.

   Facceleration=4 kg×2 m/s2=8 NFacceleration​=4kg×2m/s2=8N

Now, add the force required to overcome static friction to the force required for acceleration:
Ftotal=Fstatic+Facceleration=19.62 N+8 N=27.62 NFtotal​=Fstatic​+Facceleration​=19.62N+8N=27.62N
So, to move the box with an acceleration of 2 m/s², you need to apply a force of approximately 27.62 N parallel to the surface.