Larry stands on a bathroom scale while riding up in an elevator. His mass is 75 kg. At the current time the scale reads 100 lb. assuming the upward direction is regarded as positive, what is Larry’s acceleration in m/s2? What is happening to the elevator at this instant (i.e., is it speeding up, slowing down, or moving at a constant speed)?
To determine Larry's acceleration and the elevator's motion, we need to apply the principles of Newton's second law and relate Larry's weight to the forces acting on him. Here's how we can calculate it step by step:
1. Convert Larry's mass from kilograms (kg) to pounds (lb):
Larry's mass = 75 kg
1 kg = 2.20462 lb
Larry's mass = 75 kg * 2.20462 lb/kg = 165.35 lb
2. Determine Larry's weight (W) in newtons (N):
Weight (W) = mass (m) * gravitational acceleration (g)
The gravitational acceleration on Earth is approximately 9.8 m/s^2
Larry's weight (W) = 75 kg * 9.8 m/s^2 = 735 N
3. Compare Larry's weight with the reading on the bathroom scale:
The scale reads 100 lb, which is equivalent to 445 N (since 1 lb equals 4.45 N).
4. Analyze the forces acting on Larry:
In an elevator, three forces act on Larry: his weight (W), the normal force (N), and the net force (F_net). The normal force is the force exerted by the scale to support his weight, and the net force is the vector sum of all forces acting on him.
- W = 735 N (downward force due to gravity)
- N = 445 N (upward force exerted by the scale)
- F_net = N - W = 445 N - 735 N = -290 N (negative since it acts downward)
5. Determine Larry's acceleration using Newton's second law:
F_net = m * a
-290 N = 75 kg * a
a = -290 N / 75 kg ≈ -3.87 m/s^2
6. Interpret the negative acceleration:
The negative value of the acceleration implies that Larry is experiencing downward acceleration, which means the elevator is slowing down.
In conclusion, Larry's acceleration is approximately -3.87 m/s^2, indicating that the elevator is slowing down at this instant.