The drawing shows a 29.7-kg crate that is initially at rest. Note that the view is one looking down on the top of the crate. Two forces, F1=88N and F2=54N, are applied to the crate, and it begins to move. The angle is 55 degrees. The coefficient of kinetic friction between the crate and the floor is k = 0.324. Determine the (a) magnitude and (b) direction (relative to the x axis) of the acceleration of the crate.
To determine the magnitude and direction of the acceleration of the crate, we need to consider the forces acting on it and apply Newton's second law of motion. Let's break down the problem step by step:
Step 1: Identify the forces acting on the crate.
From the description, two forces are applied to the crate: F1 = 88N and F2 = 54N. Additionally, there is kinetic friction between the crate and the floor.
Step 2: Resolve the applied forces.
We need to resolve F1 and F2 into their x and y components. Since the angle is given as 55 degrees, we can use trigonometry to find these components:
Fx = F1 cos θ + F2 cos θ = (88N cos 55°) + (54N cos 55°)
Fy = F1 sin θ + F2 sin θ = (88N sin 55°) + (54N sin 55°)
Step 3: Calculate the frictional force.
The frictional force can be found using the formula: F_friction = μ * F_normal
where μ is the coefficient of kinetic friction and F_normal is the normal force exerted by the floor on the crate. In this case, the normal force is equal to the weight of the crate, which is given by F_normal = m * g, where m is the mass of the crate (29.7 kg) and g is the acceleration due to gravity (9.8 m/s²):
F_friction = μ * F_normal = μ * (m * g) = 0.324 * (29.7 kg * 9.8 m/s²)
Step 4: Calculate the net force.
The net force is the vector sum of all the forces acting on the crate. Given that it is initially at rest, the net force can be written as:
Net force = F_applied - F_friction = (Fx,applied + Fx,applied) - F_friction
Step 5: Apply Newton's second law of motion.
Newton's second law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Thus, we can use the formula:
Net force = mass * acceleration
Rearranging the equation, we can solve for the acceleration:
acceleration = Net force / mass
Step 6: Determine the direction of acceleration.
To determine the direction of the acceleration, we need to find the angle it makes with the x-axis. We can use the inverse tangent function to calculate this angle:
angle = arctan(Fy / Fx)
Now, let's plug in the given values and solve the problem:
1. Calculate Fx and Fy:
Fx = (88N cos 55°) + (54N cos 55°)
Fy = (88N sin 55°) + (54N sin 55°)
2. Calculate the frictional force:
F_friction = 0.324 * (29.7 kg * 9.8 m/s²)
3. Calculate the net force:
Net force = (Fx,applied + Fx,applied) - F_friction
4. Calculate the acceleration:
acceleration = Net force / mass
5. Calculate the angle of acceleration:
angle = arctan(Fy / Fx)
By following these steps, you will be able to determine the magnitude and direction of the acceleration of the crate.