A horizontal force of 150 N is used to push a 45.0-kg packing crate a distance of 5.75 m on a rough horizontal surface. If the crate moves at constant speed, find each of the following.
(a) the work done by the 150-N force
(b) the coefficient of kinetic friction between the crate and the surface
(a) 150 * 5.5 Joules
(b) no acceleration so work done by friction = 150 * 5.5
mu m g *5.5 = 150 * 5.5
mu = 150/(45*9.81)
To find each of the following, we need to use the work-energy theorem and the equation for frictional force.
(a) The work done by the 150-N force:
We know that work is given by the equation: work = force × distance × cos(theta)
Since the force and displacement are in the same direction (horizontal), the angle between them (theta) is 0 degrees.
Thus, we have:
work = 150 N × 5.75 m × cos(0°)
Since cos(0°) is equal to 1, the equation simplifies to:
work = 150 N × 5.75 m × 1
Calculating the value:
work = 862.50 J
Therefore, the work done by the 150-N force is 862.50 Joules.
(b) The coefficient of kinetic friction between the crate and the surface:
The work done against friction is equal to the work done by the applied force.
Let's assume the coefficient of kinetic friction is represented by "μk."
The work done against friction can be expressed as:
work = force of friction × distance
Using Newton's second law, we find that the force of friction is:
force of friction = μk × normal force
The normal force can be calculated using the equation:
normal force = mass × gravitational acceleration
Substituting this into the expression for the force of friction:
force of friction = μk × (mass × gravitational acceleration)
Now, we know that the work done against friction is equal to the work done by the applied force:
Force of friction × distance = applied force × distance
Substituting the known values into the equation:
μk × (mass × gravitational acceleration) × distance = 150 N × 5.75 m
Simplifying the equation:
μk = (150 N × 5.75 m) / (mass × gravitational acceleration × distance)
Substituting the given values:
μk = (150 N × 5.75 m) / (45.0 kg × 9.8 m/s^2 × 5.75 m)
Calculating the value:
μk ≈ 0.3
Therefore, the coefficient of kinetic friction between the crate and the surface is approximately 0.3.
To find the answers to these questions, we need to use several concepts from physics - namely, work and the coefficient of kinetic friction.
(a) To find the work done by the 150-N force, we use the formula:
Work = Force * Distance * cos(θ)
In this case, the force is 150 N, and the distance is 5.75 m. However, we need to determine the angle, θ, between the force and the displacement of the crate. Since the crate moves at a constant speed, we know that the net force acting on it is zero. Therefore, the force of friction must be equal in magnitude and opposite in direction to the applied force.
Since the crate is moving horizontally, the angle between the force and the displacement is 0 degrees (or cos(0) = 1). Hence, the work done by the 150-N force is:
Work = 150 N * 5.75 m * cos(0) = 150 N * 5.75 m * 1 = 862.5 J
Therefore, the work done by the 150-N force is 862.5 Joules.
(b) To find the coefficient of kinetic friction between the crate and the surface, we need to use the formula:
Force of Friction = Coefficient of Friction * Normal Force
Since the crate is on a horizontal surface and moves with a constant speed, we know that the force of friction is equal in magnitude and opposite in direction to the applied force. Hence, the force of friction is 150 N.
The normal force between the crate and the surface is the force exerted by the surface on the crate, which is equal in magnitude and opposite in direction to the gravitational force acting on the crate. The gravitational force is given by:
Force of Gravity = mass * gravity
where the mass of the crate is 45 kg, and the acceleration due to gravity is approximately 9.8 m/s^2. Therefore:
Force of Gravity = 45 kg * 9.8 m/s^2 = 441 N
Since the crate is at rest in the vertical direction, the normal force is equal to the force of gravity, which is 441 N.
Using the formula for the force of friction:
Force of Friction = Coefficient of Friction * Normal Force
we can substitute the values to get:
150 N = Coefficient of Friction * 441 N
Solving for the coefficient of friction:
Coefficient of Friction = 150 N / 441 N = 0.34 (rounded to two decimal places)
Therefore, the coefficient of kinetic friction between the crate and the surface is approximately 0.34.