A 4.5 kg block is dropped onto a spring of spring constant 1861 N/m from a height of 1500 cm find the speed of the block when the compression of the spring is 12 cm. The acceleration of gravity is 9.81 m/s
To solve this problem, we can use the principles of conservation of mechanical energy and Hooke's Law.
1. First, let's find the potential energy (PE) of the block when it is dropped from a height of 1500 cm. The potential energy can be calculated using the equation:
PE = mgh
where m is the mass of the block (4.5 kg), g is the acceleration due to gravity (9.81 m/s²), and h is the height (1500 cm = 15 m).
Substituting the values into the equation:
PE = 4.5 kg * 9.81 m/s² * 15 m
= 661.65 J
2. Next, let's find the maximum potential energy converted into the spring's potential energy (PE_spring) when the block is compressed by 12 cm. The spring's potential energy can be calculated using the equation:
PE_spring = (1/2) k x²
where k is the spring constant (1861 N/m) and x is the displacement or compression of the spring (12 cm = 0.12 m).
Substituting the values into the equation:
PE_spring = (1/2) * 1861 N/m * (0.12 m)²
= 13.3348 J
3. According to the conservation of mechanical energy, the loss in potential energy of the block (PE) is equal to the gain in the spring's potential energy (PE_spring). Therefore, we can calculate the initial kinetic energy (KE) of the block just before it reaches the spring as:
KE = PE - PE_spring
= 661.65 J - 13.3348 J
= 648.3152 J
4. Since the block is dropped, the initial kinetic energy (KE) is equal to zero.
So, the final kinetic energy (KE_final) of the block when it reaches the maximum compression of the spring can be calculated as:
KE_final = 0 J
5. We can now use the equation for kinetic energy to find the final velocity (v) of the block:
KE = (1/2) mv²
where m is the mass of the block (4.5 kg) and v is the final velocity.
Substituting the values into the equation:
0 J = (1/2) * 4.5 kg * v²
0 = 2.25 kg * v²
v² = 0 / 2.25 kg
v = 0 m/s
Therefore, the speed of the block just as it reaches the maximum compression of the spring is 0 m/s.
M = 4.5 kg
h = 1500cm = 15 m.
F = 1861N/m * 0.12m = 223.32 N.
a = F/m = 223.32/4.5 = 49.63 m/s^2
V2^2 = V1^2 + 2a*h=0 + 99.25*15=1488.8
V2 = 38.6 m/s = Speed of block.