A 350 g block is attached to a vertical spring whose stiffness constant is 12 N/m . The block is released at the position where the spring is unextended.
a) What is the maximum extension of the spring?
b)How long does it take the block to reach the lowest point?
To solve these questions, we can apply the laws of motion and Hooke's Law for springs.
a) What is the maximum extension of the spring?
To find the maximum extension of the spring, we need to consider the equilibrium position where the spring is unextended. At this position, the gravitational force acting on the block is balanced by the force exerted by the spring.
Step 1: Find the weight of the block
Given that the mass of the block is 350 g (0.35 kg), we can calculate its weight using the formula W = mg.
W = (0.35 kg) * (9.8 m/s^2) = 3.43 N
Step 2: Calculate the maximum extension
At the maximum extension, the spring force is equal to the weight of the block. According to Hooke's Law, the spring force (Fs) is given by Fs = k * x, where k is the stiffness constant of the spring and x is the extension or compression of the spring.
Since the block is released from the unextended position, the spring force equals the weight of the block. Therefore, we can set up the equation as:
k * x = W
12 N/m * x = 3.43 N
x = 3.43 N / 12 N/m
x = 0.286 m
Therefore, the maximum extension of the spring is 0.286 m.
b) How long does it take the block to reach the lowest point?
To calculate the time it takes for the block to reach the lowest point, we need to use the concept of simple harmonic motion in a vertically oscillating system.
Step 1: Find the natural frequency of the system
The natural frequency (ω) of an oscillating system is given by ω = √(k / m), where k is the stiffness constant and m is the mass.
ω = √(12 N/m / 0.35 kg)
ω = √34.28 rad/s
Step 2: Calculate the period
The period (T) of the motion is the time taken for one complete oscillation and is given by T = 2π / ω.
T = 2π / 34.28 rad/s
T ≈ 0.183 s (rounded to three decimal places)
Step 3: Calculate the time to reach the lowest point
Since the time to reach the lowest point is half the period (T/2), we can calculate it as:
Time = 0.183 s / 2
Time ≈ 0.092 s (rounded to three decimal places)
Therefore, it takes approximately 0.092 seconds for the block to reach the lowest point.