A 120-kg crate, starting from rest, is pulled across a floor with a constant horizontal force of 400 N. For the first 17 m the floor is frictionless, and for the next 17 m the coefficient of friction is 0.34. What is the final speed of the crate?
Use
F=ma to find the acceleration
and
v²-u²=2aS
to find the final velocity v from an initial velocity u over a distance of S and acceleration a.
F=400 N for the first 17m
and F=400-μmg for the next 17m, where μ is the coefficient of kinetic friction.
Post what you get for a check if you wish.
0 j
To find the final speed of the crate, we need to calculate the work done on the crate and use the work-energy principle. Here are the steps:
Step 1: Calculate the work done on the crate on the frictionless section:
The work done is given by the equation:
Work = Force * Distance
Given, Force = 400 N, Distance = 17 m
So, Work = 400 N * 17 m = 6800 N·m
Step 2: Calculate the work done against friction on the section with a coefficient of friction:
The work done against friction is given by the equation:
Work = Force of friction * Distance
The force of friction can be calculated using the equation:
Force of friction = coefficient of friction * Normal force
The normal force is equal to the weight of the crate, which is given by:
Normal force = mass * gravity
Given, coefficient of friction = 0.34, mass = 120 kg, gravity = 9.8 m/s^2, Distance = 17 m
Normal force = 120 kg * 9.8 m/s^2 = 1176 N
Force of friction = 0.34 * 1176 N = 399.84 N (approximately)
Work = 399.84 N * 17 m = 6797.28 N·m (approximately)
Step 3: Calculate the total work done on the crate:
Total Work = Work on frictionless section + Work against friction
Total Work = 6800 N·m + 6797.28 N·m = 13597.28 N·m (approximately)
Step 4: Use the work-energy principle to find the final speed of the crate:
According to the work-energy principle, the total work done is equal to the change in kinetic energy of the crate.
Change in kinetic energy = 1/2 * mass * (final velocity)^2
Given, mass = 120 kg
Let's assume the final velocity of the crate is V.
Change in kinetic energy = 1/2 * 120 kg * V^2
Equating the change in kinetic energy to the total work done:
1/2 * 120 kg * V^2 = 13597.28 N·m
Simplifying:
60 * V^2 = 13597.28
V^2 = 13597.28 / 60
V^2 ≈ 226.62
V ≈ √(226.62)
V ≈ 15.05 m/s
Therefore, the final speed of the crate is approximately 15.05 m/s.
To find the final speed of the crate, we need to calculate the work done on the crate by the applied force and the work done against friction. The net work done on an object is equal to its change in kinetic energy, which can be used to find the final speed.
First, let's calculate the work done on the crate by the applied force. The work done by a constant force is given by the formula:
Work = Force * Distance * cosθ
In this case, the applied force is 400 N and the distance is 17 m. Since the force and distance are in the same direction, the angle between them is 0°, so the cosine of 0° is 1. Therefore, the work done by the applied force is:
Work_applied = 400 N * 17 m * 1 = 6800 J
Next, let's calculate the work done against friction. The work done against friction can be calculated using the formula:
Work_friction = Force_friction * Distance
The coefficient of friction is given as 0.34, and the normal force on the crate is equal to its weight. The weight can be calculated by multiplying the mass (120 kg) by the acceleration due to gravity (9.8 m/s²):
Weight = Mass * Gravity = 120 kg * 9.8 m/s² = 1176 N
The force of friction can be calculated using the formula:
Force_friction = Coefficient of friction * normal force
Force_friction = 0.34 * 1176 N = 399.84 N
Since the force of friction acts over a distance of 17 m, the work done against friction is:
Work_friction = 399.84 N * 17 m = 6797.28 J
Now, let's calculate the net work done on the crate:
Net Work = Work_applied - Work_friction
Net Work = 6800 J - 6797.28 J = 2.72 J
Since the net work done is equal to the change in kinetic energy, we can use the formula for work to find the final velocity:
Change in Kinetic Energy = 1/2 * Mass * (Final Velocity)^2 - 1/2 * Mass * (Initial Velocity)^2
However, since the crate starts from rest, the initial velocity is 0, so the equation simplifies to:
Change in Kinetic Energy = 1/2 * Mass * (Final Velocity)^2
Plugging the values into the equation:
2.72 J = 1/2 * 120 kg * (Final Velocity)^2
Now, let's solve for the final velocity:
Final Velocity^2 = (2.72 J * 2) / (120 kg) = 0.04533 m^2/s^2
Taking the square root of both sides:
Final Velocity = √(0.04533 m^2/s^2) = 0.213 m/s
Therefore, the final speed of the crate is approximately 0.213 m/s.