A man drags a 71.5-kg crate across the floor at a constant velocity by pulling on a strap attached to the bottom of the crate. The crate is tilted 27.0° above the horizontal, and the strap is inclined 66.5° above the horizontal. The center of gravity of the crate coincides with its geometrical center. The dimensions of the crate are .9 m x .4 m. Find the magnitude of the tension in the strap.
To find the magnitude of the tension in the strap, we need to analyze the forces acting on the crate.
1. Determine the weight of the crate:
The weight, W, is given by the formula: W = m * g, where m is the mass of the crate and g is the acceleration due to gravity (9.8 m/s^2).
W = 71.5 kg * 9.8 m/s^2 = 700.7 N
2. Resolve the weight into its components:
The weight can be resolved into two components: one parallel to the incline (F_parallel) and one perpendicular to the incline (F_perpendicular).
F_parallel = W * sin(θ) = 700.7 N * sin(27°) = 296.6 N
F_perpendicular = W * cos(θ) = 700.7 N * cos(27°) = 625.8 N
3. Calculate the normal force exerted by the floor:
The normal force, N, is equal to F_perpendicular since the crate is at constant velocity.
N = F_perpendicular = 625.8 N
4. Determine the net force along the incline:
The net force along the incline can be calculated as follows:
Net force = force parallel - force friction
Since the crate is at constant velocity, the net force is zero. Therefore, force parallel = force friction.
5. Calculate the force of friction:
The force of friction, f, can be calculated using the formula f = µ * N, where µ is the coefficient of friction.
Since the crate is dragged at a constant velocity, the force of friction balances the force parallel.
Force of friction = F_parallel = 296.6 N
6. Calculate the tension in the strap:
The tension in the strap is equal to the force of friction.
Tension = Force of friction = 296.6 N
Hence, the magnitude of the tension in the strap is 296.6 N.
To find the magnitude of the tension in the strap, we can use the concept of equilibrium.
First, let's resolve the forces acting on the crate into components. We have the weight of the crate acting downward, which can be split into two components: a component perpendicular to the incline and a component parallel to the incline.
1. Calculate the weight of the crate:
The weight of an object is given by the formula: weight = mass × gravity.
Given that the mass of the crate is 71.5 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight of the crate as follows:
weight = (71.5 kg) × (9.8 m/s^2) = 700.7 N (rounded to one decimal place).
2. Resolve the weight of the crate into components:
We need to find the down the incline component and the perpendicular to the incline component.
The weight down the incline can be calculated by taking the weight of the crate and multiplying it by the sin of the angle of inclination, which is 27.0° in this case:
weight down the incline = weight × sin(27.0°).
The perpendicular to the incline component can be calculated by taking the weight of the crate and multiplying it by the cosine of the angle of inclination:
weight perpendicular to incline = weight × cos(27.0°).
3. Find the force of friction:
Since the crate is moving at a constant velocity, the frictional force opposing the motion is equal to the force parallel to the incline.
The force parallel to the incline can be calculated by taking the weight down the incline and multiplying it by the cosine of the strap angle (66.5°):
force parallel to incline = weight down the incline × cos(66.5°).
4. Find the tension in the strap:
The tension in the strap is the sum of the force parallel to the incline and the perpendicular to the incline component of the weight:
tension in the strap = force parallel to incline + weight perpendicular to incline.
5. Calculate the tension:
Now we can substitute the known values into the equations and calculate the tension in the strap.
Remember to use the appropriate units in the calculations.