A spring stretches 4.3 cm when a 13 g object is hung from it. The object is replaced with a block of mass 22 g that oscillates in simple harmonic motion, calculate the period of motion.
Period = 2*pi*sqrt(m/k)
k is the spring constant in Newtons/meter.
k = x*g/(deflection)
Use SI units (kg, m, seconds, and N/m), or else k must be in dynes/cm.
To calculate the period of motion for the block, we need to use Hooke's Law and the equation for the period of simple harmonic motion.
1. Start by calculating the spring constant (k) using Hooke's Law. Hooke's Law states that the force exerted by a spring is proportional to the displacement from its equilibrium position. The equation for Hooke's Law is given by:
F = -kx
where F is the force applied by the spring, k is the spring constant, and x is the displacement from the equilibrium position. In this case, the displacement is 4.3 cm (or 0.043 m), and the mass is 13 g (or 0.013 kg). Rearranging the equation, we have:
k = -F / x
Since F = mg (where m is the mass and g is the acceleration due to gravity), we can calculate the spring constant as:
k = -(mg) / x
Plug in the values:
k = - (0.013 kg)(9.8 m/s^2) / 0.043 m
2. Next, use the equation for the period of simple harmonic motion:
T = 2π * √(m / k)
where T is the period, m is the mass of the block (22 g or 0.022 kg), and k is the spring constant.
Plug in the values:
T = 2π * √(0.022 kg / k)
3. Substitute the value of k that we calculated earlier:
T = 2π * √(0.022 kg / - (0.013 kg)(9.8 m/s^2) / 0.043 m)
Simplifying the equation further:
T = 2π * √(0.022 kg * 0.043 m) / (-0.013 kg)(9.8 m/s^2)
Solve the equation:
T ≈ 2.87 s
Therefore, the period of motion for the block is approximately 2.87 seconds.