A force is applied to an initially stationary block of mass 1.75 kg that sits on a horizontal floor as shown. The 53.5 N force is applied at θ = 32° angle. The coefficients of friction between the floor and the block are μs = 0.593 and μk = 0.343. What is the acceleration of the block?
Henry’s solution is wrong because it fails to realize that u_s is a great enough coefficient of static friction to prevent the block from accelerating at all. Acceleration is zero
To find the acceleration of the block, we need to consider the forces acting on it. The forces involved are the gravitational force (mg), the normal force (N), and the force of friction (f).
1. Resolve the applied force into its horizontal and vertical components:
Fx = F * cos(θ)
Fy = F * sin(θ)
F = 53.5 N
θ = 32°
Fx = 53.5 N * cos(32°)
= 45.13 N (approximately)
Fy = 53.5 N * sin(32°)
= 27.79 N (approximately)
2. Calculate the normal force (N) by considering the vertical equilibrium of the block:
N - mg = 0
N = mg
m = 1.75 kg (mass of the block)
g = 9.8 m/s^2 (acceleration due to gravity)
N = 1.75 kg * 9.8 m/s^2
= 17.15 N (approximately)
3. Determine the maximum force of friction (f_max) using the equation:
f_max = μs * N
μs = 0.593 (coefficient of static friction)
N = 17.15 N
f_max = 0.593 * 17.15 N
≈ 10.17 N
4. Compare the applied horizontal force (Fx) to the maximum force of friction (f_max).
a) If Fx is less than f_max, the block remains stationary (static friction).
b) If Fx is equal to or greater than f_max, the block starts moving (kinetic friction).
Fx = 45.13 N
f_max = 10.17 N
Since Fx is greater than f_max, the block starts moving, and we need to consider kinetic friction.
5. Calculate the force of friction (f_kinetic) using the equation:
f_kinetic = μk * N
μk = 0.343 (coefficient of kinetic friction)
N = 17.15 N
f_kinetic = 0.343 * 17.15 N
≈ 5.88 N
6. Apply Newton's second law of motion to find the acceleration (a) of the block:
F - f_kinetic = ma
F = Fx (applied force in the horizontal direction)
f_kinetic = 5.88 N (force of kinetic friction)
m = 1.75 kg (mass of the block)
45.13 N - 5.88 N = 1.75 kg * a
39.25 N = 1.75 kg * a
a = 39.25 N / 1.75 kg
≈ 22.43 m/s^2 (approximately)
Therefore, the acceleration of the block is approximately 22.43 m/s^2.
To find the acceleration of the block, we need to consider the forces acting on it. In this case, we have the force applied at an angle and the force of friction.
Let's break down the problem step by step:
1. Resolve the applied force: We need to find the component of the applied force that acts parallel to the floor. Given that the force is applied at an angle of 32°, the parallel component (F_parallel) can be found by multiplying the magnitude of the force (53.5 N) by the cosine of the angle.
F_parallel = 53.5 N * cos(32°)
2. Calculate the force of friction: The coefficient of static friction (μs) is used when the object is at rest, while the coefficient of kinetic friction (μk) is used when the object is in motion. Since the block is initially stationary, we will calculate the force of static friction.
The force of static friction (F_friction) can be found by multiplying the coefficient of static friction (μs) by the normal force (F_normal). The normal force is equal to the weight of the block, which can be found by multiplying the mass (1.75 kg) by the acceleration due to gravity (9.8 m/s²).
F_normal = mass * gravity
F_normal = 1.75 kg * 9.8 m/s²
F_friction = μs * F_normal
3. Determine the net force: The net force acting on the block is the difference between the applied force (F_parallel) and the force of friction (F_friction).
F_net = F_parallel - F_friction
4. Apply Newton's second law: Newton's second law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
F_net = mass * acceleration
5. Solve for acceleration: Rearrange the equation to solve for acceleration.
acceleration = F_net / mass
Now, plug in the values we have calculated and solve for the acceleration of the block.
m*g = 1.75 * 9.8 = 17.15 N. = Wt. of block.
Fn = 17.15 + 53.5*sin32 = 45.5 N. = Normal = Force perpendicular to the floor.
Fk = u*Fn = 0.343 * 45.5 = 15.61 N. = Force of kinetic friction.
a = (Fx-Fk)/m = (53.5*Cos32-15.61)/1.75 = 17 m/s^2.