A box of books weighing 280 N is shoved across the floor of an apartment by a force of 390 N exerted downward at an angle of 35.1° 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 the time it takes to move the box 4.00 m, starting from rest, we can use the equations of motion.
First, let's break down the forces acting on the box:
1. The gravitational force acting downward, which can be calculated using the weight of the box: F_gravity = m * g
Here, m represents the mass of the box, and g is the acceleration due to gravity (approximately 9.8 m/s²).
2. The force exerted downward at an angle, which we can decompose into two components:
- The vertical component (F_vertical = F_applied * sinθ) which counteracts the gravitational force.
- The horizontal component (F_horizontal = F_applied * cosθ), which helps to push the box horizontally.
3. The frictional force opposing the motion, which can be calculated using the formula: F_friction = μ * F_normal
Here, μ is the coefficient of kinetic friction and F_normal is the normal force acting between the box and the floor.
Now, let's calculate the relevant values before we proceed:
1. Mass of the box (m):
To find the mass of the box, we can divide its weight by the acceleration due to gravity:
m = F_gravity / g
2. Normal force (F_normal):
The normal force is equal in magnitude but opposite in direction to the vertical component of the applied force:
F_normal = -F_vertical = -F_applied * sinθ
3. Frictional force (F_friction):
F_friction = μ * F_normal
Now, let's calculate the time it takes to move the box:
1. Acceleration of the box (a):
To find the acceleration, we need to calculate the net horizontal force acting on the box:
F_net_horizontal = F_horizontal - F_friction
Using Newton's second law (F = m * a), we have:
F_net_horizontal = m * a
Therefore:
a = F_net_horizontal / m
2. With the acceleration, we can find the final velocity (v_f) of the box using the equation:
v_f² = v_i² + 2 * a * d
Since the box starts from rest (v_i = 0), the equation simplifies to:
v_f = sqrt(2 * a * d)
3. Finally, we can calculate the time (t) using the equation:
t = (v_f - v_i) / a
Since the initial velocity (v_i) is 0, the equation becomes:
t = v_f / a
Now, let's plug in the values and calculate the result:
1. Calculate the mass (m):
m = F_gravity / g
m = 280 N / 9.8 m/s²
m ≈ 28.57 kg
2. Calculate the normal force (F_normal):
F_normal = -F_vertical
F_normal = -F_applied * sinθ
F_normal ≈ -390 N * sin(35.1°)
F_normal ≈ -222.47 N
3. Calculate the frictional force (F_friction):
F_friction = μ * F_normal
F_friction = 0.57 * -222.47 N
F_friction ≈ -126.92 N (negative since it opposes motion)
4. Calculate the acceleration (a):
F_net_horizontal = F_horizontal - F_friction
F_net_horizontal = F_applied * cosθ - F_friction
a = F_net_horizontal / m
a = (390 N * cos(35.1°) - (-126.92 N)) / 28.57 kg
a ≈ 6.847 m/s²
5. Calculate the final velocity (v_f):
v_f = sqrt(2 * a * d)
v_f = sqrt(2 * 6.847 m/s² * 4.00 m)
v_f ≈ 7.781 m/s
6. Calculate the time (t):
t = v_f / a
t = 7.781 m/s / 6.847 m/s²
t ≈ 1.136 s
Therefore, it takes approximately 1.136 seconds to move the box 4.00 m, starting from rest.
To find the time it takes to move the box 4.00 m, we can use the equations of motion. First, we need to calculate the net force acting on the box:
1. Resolve the force of 390 N into horizontal and vertical components:
F_horizontal = 390 N * cos(35.1°)
F_horizontal ≈ 318.47 N
F_vertical = 390 N * sin(35.1°)
F_vertical ≈ 223.48 N
2. Calculate the force of kinetic friction:
F_friction = coefficient of kinetic friction * normal force
F_friction = 0.57 * (box weight - vertical force)
F_friction = 0.57 * (280 N - 223.48 N)
F_friction ≈ 32.21 N
3. Calculate the net force acting on the box:
F_net = F_horizontal - F_friction
F_net = 318.47 N - 32.21 N
F_net ≈ 286.26 N
Now, we can use the equation of motion to find the time:
4. Apply Newton's second law:
F_net = mass * acceleration
Since the box weight (280 N) is equal to mass * gravity, we can rewrite this equation as:
F_net = weight * acceleration
Rearranging the equation, we get:
acceleration = F_net / weight
acceleration = 286.26 N / 280 N
acceleration ≈ 1.0223 m/s²
5. Use the equation of motion to find the time taken to cover a distance of 4.00 m:
s = ut + (1/2) * a * t²
Since the box starts from rest:
u = 0 m/s
Rearranging the equation, we get:
t = √((2 * s) / a)
t = √((2 * 4.00 m) / 1.0223 m/s²)
t ≈ √(7.8406 s²)
t ≈ 2.8 s
Therefore, it takes approximately 2.8 seconds to move the box 4.00 m, starting from rest.