A box of books weighing 330 N is shoved across the floor of an apartment by a force of 440 N exerted downward at an angle of 35.0° below the horizontal. If the coefficient of kinetic friction between box and floor is 0.57, how long does it take to move the box 4.00 m, starting from rest?
To find out how long it takes to move the box, we need to determine the net force acting on the box, and then use it to calculate the acceleration. Once we have the acceleration, we can apply the equations of motion to calculate the time taken.
Let's break down the forces acting on the box:
1. Weight (mg): The weight of the box pulling it downwards can be calculated using the formula weight = mass × gravitational acceleration. In this case, weight = 330 N.
2. Force pushing downwards (F): The force being exerted downward at an angle of 35.0° below the horizontal is given as 440 N.
3. Normal force (N): The normal force is equal in magnitude and opposite in direction to the weight (since the box is on a flat horizontal surface). So, the normal force is also 330 N.
4. Frictional force (f): The frictional force can be calculated using the formula frictional force = coefficient of friction × normal force. In this case, the frictional force = 0.57 × 330 N.
Now, let's determine the net force:
Net force (F_net) = F - f
Substituting the given values, we have:
F_net = 440 N - (0.57 × 330 N)
F_net = 440 N - 188.1 N
F_net = 251.9 N
Next, we can calculate the acceleration:
Using Newton's second law of motion, we have:
F_net = mass × acceleration
Solving for acceleration:
251.9 N = mass × acceleration
We need to determine the mass of the box. To find the mass, we can use the weight of the box:
weight = mass × gravitational acceleration
330 N = mass × 9.8 m/s²
Solving for mass:
mass = 330 N / 9.8 m/s²
mass ≈ 33.67 kg
Substituting the mass value back into the equation:
251.9 N = 33.67 kg × acceleration
Solving for acceleration:
acceleration = 251.9 N / 33.67 kg
acceleration ≈ 7.5 m/s²
Now, we have the acceleration. We can use the equation of motion to calculate the time taken:
Using the equation for displacement with constant acceleration:
displacement = (initial velocity × time) + (0.5 × acceleration × time²)
The box starts from rest, so the initial velocity (v₀) is 0. The equation simplifies to:
displacement = 0.5 × acceleration × time²
Substituting the given displacement value:
4.00 m = 0.5 × 7.5 m/s² × time²
Solving for time:
time² = (4.00 m) / (0.5 × 7.5 m/s²)
time² = 0.5333 s²
time ≈ √(0.5333 s²)
time ≈ 0.73 s
Therefore, it takes approximately 0.73 seconds to move the box 4.00 meters, starting from rest.
This is a somewhat involved problem.
First, you have to break the pushing force into horizontal and vertical components.
Fh=330cos35
Fv=330sin35 (goingdownward.)
Now, you have to figure friction retarding force.
frictionforce=mu(mg+downwardforce)
calculate that.
Then, you use
Net force= mass*acceleration
Fh-frictionforce= mass*acceleration
and you calcuate acceleration.
Finally, near the end, finding the time to move the box.
distance=1/2 a t^2
you know distance, and a. solve for time.