A certain mass is supported by a spring so that it is at rest 0.5 m above a table top. The lowest position of the mass is 0.1 m, and the highest position is 0.9 m. The mass is pulled down 0.4 m at time =0, creating a periodic up and down motion. It takes 1.2 s for the mass to return to the low position each time.
Determine the height of the mass above the table top after 0.7 s.
You need to know more than math to answer this question. It is about simple harmonic motion.
The amplitude of oscillation is 0.4 m and the period is 1.2 s.
The vertical height at time t is given by
y = 0.5 - 0.4 cos(2 pi t/1.2)
= 0.5 -0.4 cos(5*pi*t/3)
To answer your question, use 0.7 for t.
To determine the height of the mass above the table top after 0.7 seconds, we need to find the position of the mass at that time.
Given:
- The lowest position of the mass is 0.1 m
- The highest position of the mass is 0.9 m
- The mass is pulled down 0.4 m at time = 0
- It takes 1.2 seconds for the mass to return to the low position each time
To solve this, we can divide the total time period into equal intervals and analyze the position of the mass at each interval.
1) Calculate the total time period:
The mass takes 1.2 seconds to complete one full oscillation. Therefore, the total time period can be calculated using the formula:
Total Time Period = 2 * 1.2 seconds = 2.4 seconds
2) Divide the total time period into equal intervals:
Since the mass takes 1.2 seconds to return to the low position, the total time period can be divided into two equal intervals:
Interval 1: 0 - 1.2 seconds
Interval 2: 1.2 - 2.4 seconds
3) Determine the position of the mass in each interval:
In Interval 1 (0 - 1.2 seconds):
- At time = 0 seconds, the mass is pulled down 0.4 m, resulting in a position of 0.1 m + 0.4 m = 0.5 m above the table top.
- At time = 1.2 seconds, the mass returns to the low position of 0.1 m above the table top.
In Interval 2 (1.2 - 2.4 seconds):
- At time = 1.2 seconds, the mass is at the low position of 0.1 m above the table top.
- At time = 2.4 seconds, the mass returns to the high position of 0.9 m above the table top.
4) Calculate the position of the mass after 0.7 seconds:
The given time, 0.7 seconds, falls within Interval 1. To find the position of the mass after 0.7 seconds, we can use the linear interpolation formula:
Position = Position_1 + (Position_2 - Position_1) * (Time - Time_1) / (Time_2 - Time_1)
Where:
- Position_1 = 0.5 m (initial position at time = 0 seconds)
- Position_2 = 0.1 m (final position at time = 1.2 seconds)
- Time_1 = 0 seconds
- Time_2 = 1.2 seconds
- Time = 0.7 seconds
Substituting the values into the formula:
Position = 0.5 m + (0.1 m - 0.5 m) * (0.7 s - 0 s) / (1.2 s - 0 s)
Position = 0.5 m + (-0.4 m) * (0.7 s) / (1.2 s)
Position = 0.5 m + (-0.4 m * 0.7)
Position = 0.5 m - 0.28 m
Position = 0.22 m
Therefore, the height of the mass above the table top after 0.7 seconds is 0.22 m.
To determine the height of the mass above the table top after 0.7 seconds, we can use the equation of motion for a mass-spring system. This equation relates the displacement of the mass from its equilibrium position to the force exerted by the spring:
m * a = -k * x
where:
m = mass of the object
a = acceleration of the object
k = spring constant
x = displacement from the equilibrium position
In this case, the mass is pulled down 0.4 m at t = 0, which means the equilibrium position is 0.4 m above the table top. The lowest position is 0.1 m below the equilibrium position, and the highest position is 0.5 m above the equilibrium position.
We need to find the equation that relates the displacement from the equilibrium position to time. For a mass-spring system, the motion is usually harmonic, so we can use the equation:
x(t) = A * cos(ωt + φ)
where:
x(t) = displacement from the equilibrium position at time t
A = amplitude of the motion (maximum displacement from the equilibrium position)
ω = angular frequency of the motion (2π times the frequency of the motion)
φ = phase constant
Given that it takes 1.2 seconds for the mass to return to the low position, we can calculate the angular frequency using the formula:
ω = 2π / T
where:
T = period of the motion
In this case, T = 1.2 seconds, so:
ω = 2π / 1.2 = π / 0.6 = 5π / 3
Now, we need to determine the phase constant φ. We know that at t = 0 (when the mass is initially pulled down), the displacement is 0.4 m. Substituting t = 0 and x(t) = 0.4 into the equation, we get:
0.4 = A * cos(0 + φ)
0.4 = A * cos(φ)
To find the value of φ, we need to know the value of A (amplitude). The amplitude can be found by taking the average of the lowest and highest positions:
A = (0.5 + 0.1) / 2 = 0.6 / 2 = 0.3 m
Now we can solve for φ:
0.4 = 0.3 * cos(φ)
cos(φ) = 0.4 / 0.3
φ = arccos(0.4 / 0.3)
Using a calculator, we find φ ≈ 0.423 degrees.
Now, to determine the height of the mass above the table top after 0.7 seconds, we substitute t = 0.7 into the equation:
x(0.7) = 0.3 * cos(5π / 3 * 0.7 + 0.423)
Using a calculator, we find x(0.7) ≈ 0.3704 m.
Therefore, after 0.7 seconds, the height of the mass above the table top is approximately 0.3704 meters.