To determine the force required to move the metal box on a metal surface, we can use Newton’s second law of motion:
F=maF=ma
Where:

• FF is the force applied (in newtons, N)
• mm is the mass of the box (in kilograms, kg)
• aa is the acceleration (in meters per second squared, m/s²)

Let’s calculate the force required for each scenario:

1. Just Moving the Box (Static Friction): In this case, the box is not accelerating but is at the brink of moving. The force required to overcome static friction can be 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 (equal to the weight of the box, mgmg, where gg is the acceleration due to gravity)

   Given that μs=0.5μs​=0.5 and g=10 m/s2g=10m/s2, we can calculate NN:
   N=mg=4 kg⋅10 m/s2=40 NN=mg=4kg⋅10m/s2=40N
   Now, calculate the static friction force:
   Fstatic=0.5⋅40 N=20 NFstatic​=0.5⋅40N=20N
   So, to just move the box, you would need to apply a force of 20 N parallel to the surface.

2. Moving the Box with an Acceleration of 2 m/s²: In this case, the box is accelerating, and we need to account for both the force required to overcome static friction and the force required to accelerate the box.
   First, calculate the force required to accelerate the box:
   Facceleration=ma=4 kg⋅2 m/s2=8 NFacceleration​=ma=4kg⋅2m/s2=8N
   Now, calculate the net force required, including static friction:
   Ftotal=Fstatic+Facceleration=20 N+8 N=28 NFtotal​=Fstatic​+Facceleration​=20N+8N=28N
   So, to move the box with an acceleration of 2 m/s22m/s2, you would need to apply a force of 28 N parallel to the surface.