A woman stands on a scale in a moving elevator. Her mass is 69.1 kg, and the combined mass of the elevator and scale is an additional 734 kg. Starting from rest, the elevator accelerates upward. During the acceleration, the hoisting cable applies a force of 9430 N. What does the scale read during the acceleration?
To find out what the scale reads during the acceleration, we need to consider the forces acting on the woman in the elevator.
Let's analyze the forces in a vertical direction:
1. Weight of the woman (mg): The weight of the woman is the force exerted by gravity on her mass. The formula for weight is given by weight = mass × acceleration due to gravity. In this case, the acceleration due to gravity is approximately 9.8 m/s².
Weight of the woman (mg) = mass of the woman × acceleration due to gravity
Weight of the woman (mg) = 69.1 kg × 9.8 m/s²
2. Normal force exerted by the scale (N): The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the scale exerts an upward normal force to balance the downward weight of the woman.
3. Tension in the hoisting cable (T): The hoisting cable applies a force to lift the elevator and its contents. This force is the tension in the cable.
During the elevator's upward acceleration, there are two additional forces acting on the woman:
4. Inertial force (ma): As the elevator accelerates upwards, the woman experiences an inertial force in the opposite direction. The formula for the inertial force is given by force = mass × acceleration.
Inertial force (ma) = mass of the woman × acceleration of the elevator
Now, to find the reading on the scale, we need to calculate the net force acting on the woman. The net force is the vector sum of all the forces acting on her.
Net force = T + ma - mg
Since the elevator is moving, we know that:
Net force = mass of the woman × acceleration of the elevator
Therefore, we can simplify the equation as:
mass of the woman × acceleration of the elevator = T + ma - mg
Now, we can solve for the normal force exerted by the scale:
Normal force (N) = ma + mg
Substituting the given values:
mass of the woman = 69.1 kg
acceleration of the elevator = acceleration (not given)
To find the acceleration of the elevator, we need to use the force applied by the hoisting cable:
Net force = T + ma - mg
The net force is given by the force applied by the hoisting cable:
Net force = 9430 N
9430 = T + ma - mg
Now, we can calculate the acceleration of the elevator:
9430 = T + ma - mg
Rearranging the equation:
T = 9430 - ma + mg
Adding the weight and inertial force:
T = 9430 - (69.1 kg × acceleration) + (69.1 kg × 9.8 m/s²)
Finally, we can substitute the values into the equation for the normal force:
Normal force (N) = ma + mg = (69.1 kg × acceleration) + (69.1 kg × 9.8 m/s²)
By calculating the normal force, we can determine what the scale reads during the acceleration of the elevator.