A person has a choice while trying to move a crate across a horizontal pad of concrete: push it at a downward angle of 30 degrees, or pull it at an upward angle of 30 degrees.
If the crate has a mass of 50.0 kg and the coefficient of friction between it and the concrete is 0.750, calculate the force required to move it across the concrete at a constant speed in both situations.
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To calculate the force required to move the crate across the concrete at a constant speed, we need to consider the forces acting on the crate.
In both situations, the force required to move the crate at a constant speed is equal to the force of friction acting against it. The force of friction can be calculated using the equation:
Force of friction = coefficient of friction * normal force
The normal force is the force exerted by the surface (in this case, the concrete) perpendicular to the crate. In this scenario, the normal force is equal to the weight of the crate.
First, let's calculate the weight of the crate:
Weight = mass × acceleration due to gravity
Weight = 50.0 kg × 9.8 m/s^2
Weight = 490 N
Now, let's calculate the force of friction:
Force of friction = coefficient of friction × normal force
Force of friction = 0.750 × 490 N
Force of friction = 367.5 N
In the first situation, where the crate is pushed at a downward angle of 30 degrees, the force of friction acts against the pushing force. Therefore, the force required to move the crate can be calculated by considering the vertical and horizontal components of the force:
Vertical component of force = Force of friction × sin(angle)
Vertical component of force = 367.5 N × sin(30°)
Vertical component of force = 367.5 N × 0.5
Vertical component of force = 183.75 N
Horizontal component of force = Force of friction × cos(angle)
Horizontal component of force = 367.5 N × cos(30°)
Horizontal component of force = 367.5 N × √3/2
Horizontal component of force = 318.19 N
Therefore, when pushing the crate downward at a 30-degree angle, a force of approximately 318.19 N is required to move it across the concrete at a constant speed.
In the second situation, where the crate is pulled at an upward angle of 30 degrees, the force of friction acts in the same direction as the pulling force. Thus, the force required to move the crate is equal to the force of friction, which is 367.5 N.
Therefore, when pulling the crate upward at a 30-degree angle, a force of 367.5 N is required to move it across the concrete at a constant speed.