A 73 kg crate is dragged across a floor by pulling on a rope attached to the crate and inclined 15° above the horizontal.

(a) If the coefficient of static friction is 0.52, what minimum force magnitude is required from the rope to start the crate moving?

78867

To determine the minimum force magnitude required to start the crate moving, we need to consider the forces acting on the crate.

First, let's break down the gravitational force acting on the crate. The weight of the crate can be calculated using the formula: Weight = mass × gravity, where the mass is given as 73 kg, and gravity is approximately 9.8 m/s². Therefore, the weight of the crate is:

Weight = 73 kg × 9.8 m/s²
Weight = 715.4 N

Since the crate is inclined at an angle of 15° above the horizontal, we need to calculate the vertical and horizontal components of the weight. The vertical component can be found using the formula: Vertical Component = Weight × sin(θ), where θ is the angle of inclination (15° in this case). Substituting the values, we get:

Vertical Component = 715.4 N × sin(15°)
Vertical Component ≈ 183.14 N (rounded to two decimal places)

The horizontal component can be determined using the formula: Horizontal Component = Weight × cos(θ), where θ is the angle of inclination (15°). Substituting the values, we get:

Horizontal Component = 715.4 N × cos(15°)
Horizontal Component ≈ 686.78 N (rounded to two decimal places)

Now, let's consider the force of static friction. The force of static friction can be calculated using the formula: Force of Friction = coefficient of static friction × Normal Force, where the Normal Force is equal to the vertical component of the weight (since the crate is on a horizontal surface). Substituting the values, we get:

Force of Friction = 0.52 × 183.14 N
Force of Friction ≈ 95.25 N (rounded to two decimal places)

To start the crate moving, the minimum force magnitude required from the rope must overcome the force of static friction. Since we want to calculate the minimum force magnitude, we need to consider the force parallel to the inclined plane. This force is equal to the horizontal component of the weight plus the force of static friction. Therefore, the minimum force magnitude required to start the crate moving is:

Minimum Force Magnitude = Horizontal Component + Force of Friction
Minimum Force Magnitude ≈ 686.78 N + 95.25 N
Minimum Force Magnitude ≈ 782.03 N (rounded to two decimal places)

Therefore, a minimum force magnitude of approximately 782.03 N is required from the rope to start the crate moving.