A 0.150 kg frame is suspended from a coil spring, and stretches 0.050m. A 0.200 kg block is dropped from rest onto the fram from a height of 30.0cm. Find the maximum distance the fram moves downward from the initial position?
To find the maximum distance the frame moves downward from the initial position, we need to consider the conservation of mechanical energy.
First, let's find the potential energy stored in the spring when the frame is stretched. The potential energy stored in a spring is given by the formula:
PE = (1/2) k x^2
where PE is the potential energy, k is the spring constant, and x is the displacement of the spring from its equilibrium position.
Given that the frame stretches by 0.050 m and the mass of the frame is 0.150 kg, we can use Hooke's Law to find the spring constant. Hooke's Law states:
F = -kx
where F is the force exerted by the spring and x is the displacement from the equilibrium position.
Since the frame is in equilibrium when it is stretched, the weight of the frame is balanced by the force exerted by the spring:
mg = kx
Rearranging this equation, we can solve for k:
k = mg / x
Substituting the values (m = 0.150 kg, g = 9.8 m/s^2, x = 0.050 m), we can find the value of k.
Next, let's consider the potential energy of the block before it is dropped. The potential energy is given by:
PE = mgh
where PE is the potential energy, m is the mass of the block, g is the acceleration due to gravity, and h is the initial height of the block.
Given that the mass of the block is 0.200 kg and the initial height is 30.0 cm (converted to meters, h = 0.30 m), we can find the value of the potential energy.
Since mechanical energy is conserved, the total initial potential energy (block + spring) must be equal to the final potential energy (spring only), neglecting any energy losses due to air resistance or friction:
(PE block + PE spring)initial = PE spring final
(mgh + (1/2) k x^2)initial = (1/2) k x^2 final
Substituting the known values, we can solve for x final, the maximum distance the frame moves downward from the initial position.