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 find Larry's acceleration, we first need to convert 100 lb to Newtons (N) because the standard unit of force in the International System of Units (SI) is the Newton. The acceleration can be determined using Newton's second law of motion, which states that force is equal to mass multiplied by acceleration (F = ma).
1. Convert 100 lb to Newtons:
- 1 lb is approximately equal to 4.448 N.
- Therefore, 100 lb is equal to 100 * 4.448 = 444.8 N.
2. Apply Newton's second law of motion:
- The gravitational force acting on Larry is given by the formula F = mg, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).
- The force measured by the scale is the force exerted by Larry's mass, which includes the gravitational force and the force due to acceleration.
- So, we have F = ma + mg.
3. Plug in the values:
- F = 444.8 N (from step 1)
- m = 75 kg (given)
- 444.8 = (75 * a) + (75 * 9.8)
4. Solve for acceleration (a):
- Distribute 75 to both terms on the right side:
- 444.8 = 75a + 735
- Subtract 735 from both sides:
- 444.8 - 735 = 75a
- -290.2 = 75a
- Divide by 75 on both sides:
- a = -290.2 / 75
- a ≈ -3.8693 m/s^2
Therefore, Larry's acceleration is approximately -3.8693 m/s^2. The negative sign indicates that the elevator is moving downward or decelerating.
As for the elevator, since Larry's weight is measured as less than his actual weight, it means that the scale is exerting a force less than his gravitational force. This implies that the elevator is moving downward or decelerating at this instant.