A 22.0 kg box is sliding down on an incline set at (44) degrees up from horizontal. The coefficient of kinetic friction between the box and the incline is (.27). Find the acceleration of the box. Give your answer in m/s2 and with 3 significant figures.
mass does not matter, call it m
normal force = m g cos 44
max friction force up slope = .27 m g cos 44
gravity force down slope = m g sin 44
so
m g sin 44 - .27 m g cos 44 = m a
(sin 44 - .27 cos 44) 9.81 = a
Well, well, a sliding box and some math! Now we're really getting into the fun stuff!
Okay, let's get down to business. To find the acceleration of the box, we need to break down the forces at play here.
First, we have the force of gravity pulling the box downward, and we can calculate that by multiplying the mass of the box (22.0 kg) by the acceleration due to gravity (9.8 m/s^2). That gives us a force of about 215.6 N.
Next up, we have the normal force exerted on the box by the incline. Now, since the incline is at an angle of 44 degrees, we need to find the component of the normal force that acts parallel to the incline. This can be calculated by multiplying the normal force by the sine of the angle. The normal force is equal to the weight of the box, which we found to be 215.6 N. So, the component of the normal force parallel to the incline is about 146.2 N.
Now, we can figure out the force of friction opposing the motion. The coefficient of kinetic friction is given as 0.27, so we can multiply that by the component of the normal force parallel to the incline. This gives us a force of friction of about 39.5 N.
Finally, to find the acceleration of the box, we need to use Newton's second law, which tells us that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In this case, the net force is the force of gravity minus the force of friction, so we have:
Net Force = Force of Gravity - Force of Friction
Substituting the values we calculated earlier:
Net Force = 215.6 N - 39.5 N
Solving for the net force gives us 176.1 N. Now, we divide that by the mass of the box (22.0 kg) to find the acceleration:
Acceleration = Net Force / Mass
Acceleration = 176.1 N / 22.0 kg
Doing the math, the acceleration of the box comes out to be approximately 8.00 m/s^2.
So, there you have it! The box is accelerating at about 8.00 m/s^2. Hope that helps, and remember... physics can be a real roller coaster ride!
To find the acceleration of the box, we can start by analyzing the forces acting on it:
1. The force of gravity acting straight downward can be decomposed into two components: one parallel to the incline and one perpendicular to the incline.
The component parallel to the incline is given by: F_parallel = m * g * sin(theta), where
m = mass of the box (22.0 kg)
g = acceleration due to gravity (9.8 m/s^2)
theta = angle of the incline (44 degrees)
Plugging in the values, we get F_parallel = 22.0 kg * 9.8 m/s^2 * sin(44 degrees).
2. The force of friction acting in the opposite direction to the motion of the box is given by: F_friction = coefficient of friction * F_normal, where
F_normal is the normal force on the box.
The normal force is equal in magnitude but opposite in direction to the perpendicular component of gravity, which is given by: F_perpendicular = m * g * cos(theta), where
m = mass of the box (22.0 kg)
g = acceleration due to gravity (9.8 m/s^2)
theta = angle of the incline (44 degrees)
Therefore, F_normal = -F_perpendicular = -22.0 kg * 9.8 m/s^2 * cos(44 degrees).
Plugging in the values, we get F_friction = 0.27 * (-22.0 kg * 9.8 m/s^2 * cos(44 degrees)).
3. The net force acting on the box is the sum of the parallel component of gravity and the force of friction.
Net force = F_parallel + F_friction.
4. The acceleration of the box can be calculated using Newton's second law: F_net = m * a, where
m = mass of the box (22.0 kg)
a = acceleration of the box.
Therefore, F_net = 22.0 kg * a.
Set this equal to the net force: F_net = F_parallel + F_friction.
5. Finally, solve for the acceleration, a.
22.0 kg * a = 22.0 kg * 9.8 m/s^2 * sin(44 degrees) + 0.27 * (-22.0 kg * 9.8 m/s^2 * cos(44 degrees)).
Simplify and solve for a.
By performing the calculations, the acceleration of the box is approximately -3.64 m/s^2 (including the correct significant figures). Note that the negative sign indicates that the acceleration is in the opposite direction of the motion (up the incline).
To find the acceleration of the box, we can use Newton's second law of motion, which states that the net force acting on an object is equal to its mass times acceleration (F_net = m * a).
First, we need to calculate the net force on the box. There are two forces acting on the box: the gravitational force and the force of kinetic friction.
1. Calculate the gravitational force (F_gravity):
The gravitational force is determined by the weight of the box. The weight (W) is given by W = m * g, where m is the mass of the box (22.0 kg) and g is the acceleration due to gravity (9.8 m/s^2).
W = 22.0 kg * 9.8 m/s^2 = 215.6 N
2. Calculate the force of kinetic friction (F_friction):
The force of kinetic friction is given by F_friction = coefficient of kinetic friction * normal force. The normal force (F_normal) is the component of the weight of the box that is perpendicular to the incline and can be calculated as F_normal = W * cos(theta), where theta is the angle of the incline (44 degrees).
F_normal = 215.6 N * cos(44 degrees) = 153.3 N
F_friction = 0.27 * 153.3 N = 41.4 N
3. Calculate the net force (F_net):
The net force is determined by subtracting the force of friction from the gravitational force.
F_net = F_gravity - F_friction
F_net = 215.6 N - 41.4 N = 174.2 N
Now, we can plug the net force into Newton's second law equation to find the acceleration (a):
F_net = m * a
174.2 N = 22.0 kg * a
Divide both sides of the equation by 22.0 kg:
a = 174.2 N / 22.0 kg = 7.918182 m/s^2
Therefore, the acceleration of the box is approximately 7.92 m/s^2.