A spring is resting vertically on a table. A small box is dropped onto the top of the spring and compresses it. Suppose the spring has a spring constant of 450 N/m and the box has a mass of 1.5 kg. The speed of the box just before it makes contact with the spring is 0.49 m/s.
(b) What is the magnitude of the spring's displacement when the spring is fully compressed?
To find the magnitude of the spring's displacement when it is fully compressed, we need to use the principle of conservation of mechanical energy.
The total mechanical energy before the box makes contact with the spring is given by the formula:
E_initial = KE_initial + PE_initial
Since the box is dropping vertically, it has only kinetic energy (KE_initial) just before it makes contact with the spring. The potential energy (PE_initial) is zero since the spring is not yet compressed.
The total mechanical energy when the spring is fully compressed is given by:
E_final = KE_final + PE_final
Since the box comes to rest at the maximum compression, it has zero kinetic energy (KE_final). The potential energy (PE_final) is maximum when the spring is fully compressed.
The formula for potential energy of a spring is:
PE = (1/2)kx^2
Where:
PE = potential energy
k = spring constant
x = displacement of the spring
Setting up the conservation of mechanical energy equation:
KE_initial + PE_initial = KE_final + PE_final
Since KE_initial is (1/2)mv^2 (where m is the mass of the box and v is its velocity just before it makes contact with the spring), and PE_initial is zero, the equation becomes:
(1/2)mv^2 = (1/2)kx^2
Rearranging the equation to solve for x (the displacement of the spring):
x^2 = (mv^2) / k
x = sqrt((mv^2) / k)
Plugging in the given values, we have:
m = 1.5 kg
v = 0.49 m/s
k = 450 N/m
x = sqrt((1.5 kg * (0.49 m/s)^2) / 450 N/m)
x ≈ 0.0525 m
Therefore, the magnitude of the spring's displacement when it is fully compressed is approximately 0.0525 meters.
To find the magnitude of the spring's displacement when it is fully compressed, we can use the principle of conservation of mechanical energy. The initial potential energy of the box is equal to the final potential energy of the spring when it is fully compressed.
The potential energy of the box can be calculated using the formula:
Potential Energy (PE) = mass (m) * gravity (g) * height (h)
Given that the mass of the box is 1.5 kg and the acceleration due to gravity is approximately 9.8 m/s^2, the potential energy of the box just before it makes contact with the spring can be calculated as follows:
PE_box_initial = m * g * h
To find the height (h) of the box above the spring, we can use the fact that its initial speed is 0.49 m/s and the final speed will be zero when it is fully compressed. Using the equations of linear motion, we have:
Final Velocity (Vf) = Initial Velocity (Vi) + (2 * acceleration * displacement)
Since the final velocity is zero, the equation becomes:
0 = 0.49 m/s + (2 * (-9.8 m/s^2) * displacement)
Simplifying the equation, we get:
0.49 m/s = -19.6 m/s^2 * displacement
Solving for displacement, we find:
displacement = 0.49 m/s / (-19.6 m/s^2) = -0.025 m
Since the displacement is negative, it indicates that the spring is compressed.
To find the potential energy of the fully compressed spring, we can use the formula:
PE_spring_final = 0.5 * k * (displacement)^2
Given that the spring constant is 450 N/m and the displacement is -0.025 m, we can calculate the potential energy as follows:
PE_spring_final = 0.5 * 450 N/m * (-0.025 m)^2 = 0.28125 J
Therefore, the magnitude of the spring's displacement when it is fully compressed is approximately 0.025 m.
F = mg - kx
ma = mg -kx
when a equals zero
kx = mg
x=mg/k
Plug and chug your numbers