Zach, whose mass is 67 kg, is in an elevator descending at 12 m/s. The elevator takes 3.5 s to brake to a stop at the first floor.
(a) What is Zach's apparent weight before the elevator starts braking?
(b) What is Zach's apparent weight while the elevator is braking?
a) weight= mg+ma=m(g+0)
b) weight= m(g+12/3.5)
(a) Zach's apparent weight before the elevator starts braking is the same as his actual weight, which is 67 kg multiplied by the acceleration due to gravity, which is approximately 9.8 m/s². So his apparent weight is 67 kg × 9.8 m/s² = 656.6 N.
(b) During the braking, the elevator is decelerating at a rate that will bring it to a stop in 3.5 s. To find Zach's apparent weight, we need to find the net force acting on him.
Using the equation F = ma, where F is the net force, m is Zach's mass, and a is the acceleration, we can rearrange the equation to find the net force:
F = m × a
Since a = Δv/Δt (change in velocity divided by the change in time), we can calculate the net force:
F = m × (Δv/Δt)
Δv is the change in velocity. Before the elevator starts braking, Zach is descending at a velocity of 12 m/s. After the elevator stops, his velocity will be 0 m/s. So the change in velocity is:
Δv = 0 m/s - (-12 m/s) = 12 m/s
Plugging this into the equation, we have:
F = 67 kg × (12 m/s) / 3.5 s
Calculating that, we get:
F ≈ 230.29 N
So Zach's apparent weight while the elevator is braking is approximately 230.29 N.
To solve this problem, we'll need to use Newton's second law (F=ma) and the equation for acceleration (a = (v - u) / t), where F is the force, m is the mass, a is the acceleration, v is the final velocity, u is the initial velocity, and t is the time.
Given:
Zach's mass (m) = 67 kg
Initial velocity (u) = 0 m/s (since the elevator is not yet moving or braking)
Final velocity (v) = -12 m/s (negative sign denotes downward direction)
Time taken to brake (t) = 3.5 s
Let's calculate the answers to each part of the question:
(a) What is Zach's apparent weight before the elevator starts braking?
When the elevator is descending at a constant velocity of 12 m/s, its acceleration is 0 m/s^2, and Zach's weight is equal to his actual weight.
Zach's apparent weight before the elevator starts braking = Zach's actual weight
= m * g (where g is the acceleration due to gravity)
= 67 kg * 9.8 m/s^2 (approximating g as 9.8 m/s^2)
= 656.6 N
Therefore, Zach's apparent weight before the elevator starts braking is approximately 656.6 N.
(b) What is Zach's apparent weight while the elevator is braking?
To find Zach's apparent weight while the elevator is braking, we need to calculate the net force acting on him. The net force is the force required to decelerate the elevator, which can be calculated using Newton's second law.
Using the equation a = (v - u) / t, we can find the acceleration of the elevator:
a = (-12 m/s - 0 m/s) / 3.5 s
= -12 m/s / 3.5 s
= -3.43 m/s^2
Now we can calculate the net force acting on Zach:
Net force (F) = m * a
= 67 kg * -3.43 m/s^2
= -230.81 N (negative sign indicates upward force)
The apparent weight is the force experienced by Zach, so:
Zach's apparent weight while the elevator is braking = Zach's actual weight + Net force
= m * g + F
= 67 kg * 9.8 m/s^2 + (-230.81 N)
= 656.6 N - 230.81 N
= 425.79 N
Therefore, Zach's apparent weight while the elevator is braking is approximately 425.79 N.
To find Zach's apparent weight in each of the given scenarios, we need to consider the forces acting on him and use the concept of Newton's second law of motion, which states that force equals mass multiplied by acceleration.
(a) Zach's apparent weight before the elevator starts braking can be determined using the following steps:
Step 1: Determine the net force acting on Zach before the elevator starts braking.
In this scenario, the only force acting on Zach is the gravitational force (weight). The formula for weight is given by:
Weight (W) = mass (m) × acceleration due to gravity (g)
where acceleration due to gravity is approximately 9.8 m/s².
Given that Zach's mass is 67 kg, we can calculate his weight:
W = 67 kg × 9.8 m/s²
W ≈ 657.6 N
So, the net force on Zach before the elevator starts braking is approximately 657.6 N.
Step 2: Determine Zach's apparent weight.
Zach's apparent weight is equal to the net force acting on him. Therefore, his apparent weight before the elevator starts braking is approximately 657.6 N.
(b) Zach's apparent weight while the elevator is braking can be determined using the following steps:
Step 1: Identify the forces acting on Zach while the elevator is braking.
During the elevator's braking, two forces act on Zach:
1. The gravitational force (weight) in the downward direction.
2. The upward force exerted on Zach by the elevator to counteract his weight.
Step 2: Determine the net force acting on Zach while the elevator is braking.
The net force acting on Zach is the difference between the upward force exerted by the elevator and his weight.
Considering the upward force exerted by the elevator as positive and Zach's weight as negative, we can express the net force as:
Net force = upward force - weight
Step 3: Calculate the net force.
To calculate the net force, we need to find the magnitude of the upward force exerted by the elevator. This can be determined using the equation:
Net force = mass × acceleration
Given that the elevator's acceleration is equal to the negative rate of its velocity change, we can calculate the acceleration:
Acceleration = change in velocity / time
The elevator's velocity changes from 12 m/s to 0 m/s in a time of 3.5 s. Therefore:
Acceleration = (0 - 12) m/s / 3.5 s
Acceleration ≈ -3.43 m/s² (negative because it's in the opposite direction to the initial velocity)
Step 4: Substitute the values into the equation for the net force:
Net force = mass × acceleration
Net force = 67 kg × (-3.43 m/s²)
Net force ≈ -230.81 N
So, the net force acting on Zach while the elevator is braking is approximately -230.81 N.
Step 5: Determine Zach's apparent weight.
Zach's apparent weight is equal to the magnitude of the net force acting on him. Therefore, his apparent weight while the elevator is braking is approximately 230.81 N.