Askiitians Tutor Team
Last Activity: 4 Months ago
To solve the problem of a block attached to a spring oscillating after the platform is removed, we need to analyze the motion of the block and the properties of the spring. Let's break this down step by step.
Understanding the System
When the platform is removed, the block attached to the spring will experience a downward force due to gravity, causing it to oscillate. The spring will exert an upward force as it stretches. The point where the block reaches its lowest position (5 cm below the initial position) is known as the equilibrium position of the oscillation.
Key Concepts
- Simple Harmonic Motion (SHM): The motion of the block can be modeled as SHM, where the restoring force is proportional to the displacement from the equilibrium position.
- Amplitude: The maximum displacement from the equilibrium position. In this case, the amplitude is 5 cm.
- Angular Frequency (ω): This is given by the formula ω = √(k/m), where k is the spring constant and m is the mass of the block.
Calculating Time Period and Frequency
The time period (T) of the oscillation can be calculated using the formula:
T = 2π√(m/k)
However, we need to find the time taken to reach a height of 3 cm from the lowest position (which is 5 cm below the initial position). This means we are looking for the time taken to move from 5 cm to 2 cm below the initial position.
Finding the Displacement
In terms of displacement from the equilibrium position, moving from 5 cm to 2 cm below the initial position means the block moves up 3 cm. In SHM, the displacement can be expressed as:
x(t) = A cos(ωt)
Where A is the amplitude (5 cm) and x(t) is the displacement at time t. We want to find the time when:
x(t) = -2 cm (since the lowest position is -5 cm and we want to find when it is at -2 cm).
Setting Up the Equation
We can set up the equation:
-2 = 5 cos(ωt)
Solving for cos(ωt):
cos(ωt) = -2/5
Finding the Angle
Now, we need to find the angle ωt:
ωt = cos⁻¹(-2/5)
To find ω, we need the spring constant k. If we assume we have k, we can calculate ω and then find the time t.
Final Calculation
Once we have ω, we can calculate t:
t = (1/ω) * cos⁻¹(-2/5)
Keep in mind that the cosine function will yield two angles in the range of 0 to 2π, and we are interested in the first instance where the block reaches that position as it moves upwards.
Example Calculation
For example, if we assume k = 100 N/m and m = 1 kg, we can find:
ω = √(k/m) = √(100/1) = 10 rad/s
Then, we can calculate:
t = (1/10) * cos⁻¹(-2/5)
Using a calculator, cos⁻¹(-0.4) gives approximately 2.214 rad. Thus:
t ≈ 0.2214 seconds.
In summary, the time taken for the block to reach a height of 3 cm from its lowest position can be calculated using the principles of SHM, provided we know the spring constant and mass. This approach allows us to understand the oscillatory motion of the block effectively.
