A metal box of mass 4kg rests on the top of a metal surface. What force applied parallel to the surface is required to (i) just move the box? (ii) move the box with an acceleration of 2m/s2?
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StopLearn Team Staff answered 8 months ago 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.
- 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.
- 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.
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