A shoe case weighing 40 lb is located on a scale inside a moving elevator that is going upward with the speed of 8 m/s. At time T1 elevator slows down with the acceleration of 3 m/s2 until the time T2. What are the the readings of the scale at T2 and T1 respectively?
While the upward velocity is decreasing, there is downward acceleration at rate a = 3 m/s^2
At time T1, the scale reads force F.
W - F = M a
F = W - M a = W - (W/g) a
= W (1 - a/g) = 40 lb * (6.8/9.8)
If acceleration = 0 at T2, the scale reads 40 lb then. We are able to use metric units for a and g because We are calculating a ratio.
To determine the readings of the scale at times T2 and T1, we need to consider the forces acting on the shoe case in both situations.
At T2 (when the elevator has slowed down):
1. Find the net force acting on the shoe case.
- The force due to gravity (weight) is given by Fg = m * g, where m is the mass of the shoe case and g is the acceleration due to gravity (approximately 9.8 m/s^2).
- In this case, the weight (Fg) is 40 lb. To convert it to mass, we use the conversion factor: 1 lb = 0.4536 kg.
- So, the mass of the shoe case is approximately m = 40 lb * 0.4536 kg/lb = 18.14 kg.
- The net force is the difference between the force due to gravity and the force due to acceleration: F_net = Fg - F_acc.
- Since the elevator is slowing down, the acceleration will oppose the force due to gravity, so F_acc = m * (-a).
- Plugging in the values, we get F_net = Fg - F_acc = m * g - m * (-a).
2. Calculate the scale reading.
- The scale measures the force acting on the shoe case, which is equal to the net force.
- Therefore, the scale reading at T2 will be equal to F_net.
At T1 (when the elevator is moving upward):
1. Find the net force acting on the shoe case.
- The force due to gravity (weight) remains the same, Fg = m * g.
- In addition to the force due to gravity, we need to consider the acceleration of the elevator, which is opposing its motion.
- The net force is now the sum of the force due to gravity and the force due to the acceleration: F_net = Fg + F_acc.
- Since the elevator is moving upward, the acceleration will assist the force due to gravity, so F_acc = m * a.
- Plugging in the values, we get F_net = Fg + F_acc = m * g + m * a.
2. Calculate the scale reading.
- Similar to the previous case, the scale reading at T1 will be equal to F_net.
Now that we have the mathematical expressions for the net force and the scale readings at T2 and T1, we can calculate the values using the given information.