A 25kg box is pulled across a frictionless floor by a rope that is inclined 30 degrees above horizontal. The tension in the rope is 50N. How much work is done by the force in moving the crate 10 meters? If the box started at rest, what is the final velocity of the box?
A 50Kg box slid across the floor at a constant velocity of 20m/s, with a 45N force acting at 40o with respect to the horizontal. The coefficient of kinetic friction between the box and the surface is 0.3. Find,
a) The frictional force acting on the box.
b) The normal force
c) The acceleration of the box in 25s
d) Distance covered in 30s
e) The kinetic energy
f) The work done.
To determine the work done by the force and the final velocity of the box, we can break down the problem into two parts: finding the work done and finding the final velocity.
1. Work Done:
The work done by a force is given by the product of the magnitude of the force, the displacement, and the cosine of the angle between the force and the displacement. In this case, the force is the tension in the rope, which is 50N, and the displacement is 10 meters. The angle between the force and displacement is the angle of inclination, which is 30 degrees. Therefore, we have:
Work = Force * Displacement * cos(θ)
Substituting the given values, we get:
Work = 50N * 10m * cos(30°)
To calculate the cosine of 30 degrees, we can use a scientific calculator or lookup table. The cosine of 30 degrees is approximately 0.866.
Work = 50N * 10m * 0.866
Now we can calculate the value:
Work = 433 J
Therefore, the work done by the force in moving the box is 433 Joules.
2. Final Velocity:
To find the final velocity of the box, we can use the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy. Mathematically, this can be expressed as:
Work = ΔKE
Since no initial velocity is given, the initial kinetic energy is zero. Therefore, the equation becomes:
Work = KE_final - KE_initial
From the equation above, we can see that the work done is equal to the change in kinetic energy. So, the work done by the force is equal to the final kinetic energy of the box.
Work = KE_final
Using the work done value we calculated earlier:
433 J = KE_final
To find the final kinetic energy, we need to use the formula for kinetic energy:
KE = (1/2) * m * v^2
where m is the mass of the box, and v is the final velocity. The given mass is 25 kg. Substituting the values, we have:
433 J = (1/2) * 25 kg * v^2
Rearranging the equation to solve for v:
v^2 = (433 J * 2) / 25 kg
v^2 = 17.32 m^2/s^2
v = √17.32 m/s
v ≈ 4.16 m/s
Therefore, the final velocity of the box is approximately 4.16 m/s.