A group of construction workers are building a house and want to lower down an excess bundle of 1420 kg bricks to ground. They tie one end of a rope to the bundle of bricks, loop it through a pulley, and tie the other end of the rope to a toolbox. If the coefficient of kinetic friction between the toolbox and the floor is 0.690 and the acceleration of the bundle of bricks is 6.20 m/s^2, what is the mass of the toolbox?
To find the mass of the toolbox, we need to consider the forces acting on the system.
The two main forces involved are the force of tension in the rope and the force of friction between the toolbox and the floor. Let's break down these forces:
1. The force of tension in the rope: The rope is tied to both the bundle of bricks and the toolbox. Since the bundle of bricks is accelerating downward, there must be a net force pulling the bundle downward. This force is provided by the tension in the rope. Let's call this force T.
2. The force of friction between the toolbox and the floor: As the bundle of bricks accelerates downward, it causes a tension force on the rope, which in turn causes a horizontal force on the toolbox. This horizontal force causes a frictional force between the toolbox and the floor. Let's call this force f (frictional force).
Now, let's analyze the forces involved in this scenario.
1. Force of tension (T): This force is responsible for accelerating the bundle of bricks downward. The magnitude of this force is equal to the weight of the bundle of bricks (W = m_bundle * g), where m_bundle is the mass of the bundle of bricks and g is the acceleration due to gravity (approximately 9.8 m/s^2). Therefore, T = m_bundle * g.
2. Force of friction (f): The frictional force is proportional to the normal force between the toolbox and the floor. The normal force is equal to the weight of the toolbox (W = m_toolbox * g), where m_toolbox is the mass of the toolbox. Therefore, f = μ * N, where μ is the coefficient of kinetic friction and N is the normal force (N = m_toolbox * g). Therefore, f = μ * m_toolbox * g.
According to Newton's second law, the net force on an object is equal to its mass multiplied by its acceleration. Taking the downward direction as positive, we have:
T - f = m_bundle * a
Substituting the values we have:
m_bundle * g - μ * m_toolbox * g = m_bundle * a
Simplifying this expression:
m_bundle * (g - a) = μ * m_toolbox * g
Now, we can solve for m_toolbox:
m_toolbox = (m_bundle * (g - a)) / (μ * g)
Plugging in the given values:
m_toolbox = (1420 kg * (9.8 m/s^2 - 6.20 m/s^2)) / (0.690 * 9.8 m/s^2)
By evaluating this expression, we can find the mass of the toolbox.