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 24.5° above the horizontal, and the strap is inclined 63.0° above the horizontal. The center of gravity of the crate coincides with its geometrical center, as indicated in the drawing. Find the magnitude of the tension in the strap.
To find the magnitude of the tension in the strap, we can use the concept of equilibrium. In order for the crate to move at a constant velocity, the sum of all the forces acting on it must be zero.
Let's break down the forces acting on the crate:
1. Weight (downward force): The weight of the crate can be calculated by multiplying the mass (71.5 kg) by the acceleration due to gravity (9.8 m/s²). The weight acts vertically downward.
2. Normal force (upward force): The normal force is the force exerted by the floor on the crate and acts perpendicular to the floor. In this case, because the crate is tilted, the normal force can be resolved into two components: one that is perpendicular to the floor and the other that is parallel to the floor.
3. Tension in the strap: This is the force that the person is pulling with and acts along the direction of the strap.
Since the crate is moving at a constant velocity, there is no acceleration in either the vertical or horizontal direction. Therefore, the sum of the vertical forces and the sum of the horizontal forces must be zero.
First, let's calculate the weight of the crate:
Weight = mass * acceleration due to gravity
Weight = 71.5 kg * 9.8 m/s²
Next, let's resolve the weight into components based on the angle of the crate:
Vertical component of weight = Weight * sin(24.5°)
Horizontal component of weight = Weight * cos(24.5°)
Now, considering the equilibrium conditions:
Sum of vertical forces = 0:
The vertical component of the weight (acting downwards) is balanced by the perpendicular component of the normal force (acting upwards):
Vertical component of weight = Perpendicular component of normal force
Sum of horizontal forces = 0:
The horizontal component of the weight (acting to the left) is compensated by the tension in the strap (acting to the right):
Horizontal component of weight = Tension in the strap * cos(63.0°)
Now you can substitute the calculated and given values into the equations and solve for the tension in the strap.