Movers are looking to lower a cabinet out of an open window to a mattress on the ground below. One end of a rope is tied to a 188 kg dresser, looped through a pulley, and the other end of the rope is tied to a cabinet. If the coefficient of kinetic friction between the dresser and floor is 0.410 and the tension on the rope is 1.23×103 N, what is the mass of the cabinet?
To solve this problem, we can use the principles of Newton's laws of motion and the equations of motion. Here's how we can approach it:
1. First, let's define the forces acting on the system. We have the gravitational force acting on both the dresser and the cabinet, and the force of kinetic friction acting on the dresser.
2. According to Newton's second law of motion, the net force acting on an object is equal to the product of its mass and acceleration. Since the dresser is not accelerating vertically, the net force acting on it in the vertical direction must be zero. Therefore, the gravitational force acting downward on the dresser must be balanced by the tension in the rope:
Tension = Weight of the dresser (m_1 * g)
Here, m_1 represents the mass of the dresser and g is the acceleration due to gravity (approximately 9.8 m/s^2).
3. Now, let's consider the horizontal motion of the dresser. The force of kinetic friction can be calculated using the formula:
Force of kinetic friction = Coefficient of kinetic friction * Normal force
In this case, the normal force is equal to the weight of the dresser since it is on a horizontal surface. Therefore,
Force of kinetic friction = Coefficient of kinetic friction * Weight of the dresser
4. Since there is no acceleration in the horizontal direction, the net force acting on the dresser in that direction must also be zero. Therefore,
Net force in the horizontal direction = Tension - Force of kinetic friction
Setting the net force equal to zero, we have:
Tension - Force of kinetic friction = 0
Substituting the expressions for tension and force of kinetic friction, we can then solve for the mass of the dresser.
5. Once we have the mass of the dresser, we can use the relationship described at the beginning of the problem to find the mass of the cabinet. From the given information, the tension in the rope is also equal to the weight of the cabinet (m_2 * g):
Tension = Weight of the cabinet (m_2 * g)
Rearranging this equation, we can solve for the mass of the cabinet.
By following this step-by-step approach, you should be able to find the mass of the cabinet in this scenario.