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 193 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 0.680 and the tension on the rope is 2.39×103 N, what is the mass of the cabinet?
I wonder if the cabinet is accelerating down
call it a
ma = mg - T
m(9.81-a) = 2390
T - .68 * 193 *9.81 = 193 a
or
193 a = 2390 - 1287
a = 5.71 m/s^2 (dangerous)
m (9.81-5.71) = 2390
m = 583 kg
To find the mass of the cabinet, we need to analyze the forces acting on the system and make use of Newton's laws of motion.
Let's break down the problem step by step:
1. Identify the forces:
- The tension in the rope: T
- The force of gravity acting on the dresser: F_g_dresser
- The force of gravity acting on the cabinet: F_g_cabinet
- The kinetic friction force between the dresser and the floor: F_friction
2. Resolve the forces:
- The tension in the rope is equal to the force of gravity acting on the dresser:
T = F_g_dresser
- The force of gravity acting on the dresser can be calculated using the mass and gravitational acceleration:
F_g_dresser = m_dresser * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- The force of gravity acting on the cabinet can also be calculated using its mass and gravitational acceleration:
F_g_cabinet = m_cabinet * g
- The kinetic friction force between the dresser and the floor can be calculated using the coefficient of kinetic friction:
F_friction = coefficient_of_friction * F_n, where F_n is the normal force.
3. Analyze the forces:
- The normal force acting on the dresser is equal to its weight since it is resting on a horizontal surface:
F_n = F_g_dresser
- The normal force acting on the cabinet is also equal to its weight:
F_n = F_g_cabinet
4. Set up the equations:
- From step 2, we know that T = F_g_dresser and F_friction = coefficient_of_friction * F_n
- Substituting F_g_dresser for T in F_friction, we get F_friction = coefficient_of_friction * T
5. Solve for the mass of the cabinet (m_cabinet):
- By re-arranging the equation F_friction = coefficient_of_friction * T and substituting F_n = m_cabinet * g, we get:
coefficient_of_friction * T = m_cabinet * g
- Solving for m_cabinet, we have:
m_cabinet = (coefficient_of_friction * T) / g
Now, we can plug in the given values and calculate the mass of the cabinet:
coefficient_of_friction = 0.680
T = 2.39 × 10^3 N
g = 9.8 m/s^2
m_cabinet = (0.680 * 2.39 × 10^3) / 9.8
Calculating this expression, we find that the mass of the cabinet is approximately 165 kg.
To find the mass of the cabinet, we can begin by analyzing the forces acting on the system.
Let's denote the mass of the cabinet as M.
1. Weight of the dresser:
The weight of the dresser is given by:
Weight = mass × acceleration due to gravity = 193 kg × 9.8 m/s² = 1893.4 N.
2. Tension force:
The tension force in the rope is given as 2.39 × 10³ N, acting upwards.
3. Frictional force on the dresser:
The frictional force can be calculated using the coefficient of kinetic friction and the normal force.
Frictional force = coefficient of kinetic friction × normal force.
Since the dresser is moving downwards, the normal force is equal to the weight of the cabinet (1893.4 N).
Frictional force = 0.680 × 1893.4 N = 1289.4 N (approximately).
Now, let's consider the forces acting on the cabinet.
4. Weight of the cabinet:
The weight of the cabinet is given by:
Weight = mass × acceleration due to gravity = M × 9.8 m/s².
5. Tension force:
The tension force in the rope is the same as in the case of the dresser, which is 2.39 × 10³ N.
6. Frictional force on the cabinet:
The frictional force can again be calculated using the coefficient of kinetic friction and the normal force.
Frictional force = coefficient of kinetic friction × normal force.
Since the cabinet is stationary on the floor, the normal force is equal to the weight of the cabinet (M × 9.8 m/s²).
Frictional force = 0.680 × (M × 9.8 m/s²) = 6.764 M N (approximately).
Now, let's set up the equation of equilibrium in the vertical direction:
Tension force - Weight of the dresser - Frictional force on the dresser - Weight of the cabinet + Tension force - Frictional force on the cabinet = 0.
Substituting the values we calculated before:
2.39 × 10³ N - 1893.4 N - 1289.4 N - (M × 9.8 m/s²) + 2.39 × 10³ N - 6.764 M N = 0.
Combining like terms:
-1893.4 N - 1289.4 N + 2.39 × 10³ N + 2.39 × 10³ N = (M × 9.8 m/s²) + 6.764 M N.
Simplifying further:
907.2 N = 16.564 M N.
Dividing both sides by 16.564 M N:
907.2 N / 16.564 M N = M.
Approximately:
54.836 kg = M.
Therefore, the mass of the cabinet is approximately 54.836 kg.