A 4kg box is compressed 50cm on a spring (k=800 N/m) and then slides across a horizontal floor. If (u=0.4) between the box and the floor...
A) how much work is done?
B) assuming the frictional force is constant, how far will the box slide before it comes to a stop?
A)
The elastic potential energy stored in the spring would convert into KE of the box. This KE would be dissipated while working against the frictional force.
So work done = (1/2)Kx^2
= 800*0.5^2/2
= 100 Joules
B)
Frictional force F= mu*mg
F = 0.4*4*10 = 16 N
Work done = F*d
d = 100/16 = 6.25 m
To find the answers to these questions, we need to break down the problem into different parts. Let's start with the first question.
A) How much work is done?
The work done on an object can be calculated using the formula:
Work = Force × Distance × cos(θ)
In this case, the force responsible for the work is the force exerted by the spring. The distance is the compression of the spring, and θ is the angle between the displacement and the force vectors (which is 0 degrees since they are in the same direction).
The force exerted by the spring can be calculated using Hooke's Law:
Force = k × displacement
Substituting the given values, we can calculate the work done:
Force = 800 N/m × 0.5 m
Force = 400 N
Work = 400 N × 0.5 m × cos(0°)
Work = 400 N × 0.5 m × 1
Work = 200 J
Therefore, the work done on the box is 200 Joules.
B) Assuming the frictional force is constant, how far will the box slide before it comes to a stop?
To find the distance the box will slide, we can use the work-energy principle. According to this principle, the work done by all the forces acting on an object is equal to the change in its kinetic energy.
In this case, the work done is the work due to friction, given by the equation:
Work_friction = Force_friction × Distance
The force of friction can be calculated using the equation:
Force_friction = μ × Normal force
The normal force is equal to the weight of the box in this case:
Normal force = mass × gravitational acceleration
Normal force = 4 kg × 9.8 m/s^2
Normal force = 39.2 N
Substituting the given coefficients, we can calculate the force of friction:
Force_friction = 0.4 × 39.2 N
Force_friction = 15.68 N
Now, we can use the work-energy principle to find the distance the box will slide:
Work_friction = -ΔKinetic energy
Since the box comes to a stop, the change in kinetic energy is equal to its initial kinetic energy.
Work_friction = -0.5 × mass × initial velocity^2
Substituting the given mass and assuming the initial velocity is 0:
Work_friction = -0.5 × 4 kg × (0 m/s)^2
Work_friction = 0 J
Therefore, the work done by friction is 0 J.
Now, we can set the work done by friction equal to the work due to friction and find the distance:
Work_friction = -Force_friction × Distance
0 J = -15.68 N × Distance
Solving for Distance:
Distance = 0 J / -15.68 N
Distance = 0 m
Therefore, the box will not slide any distance before coming to a stop.