Question

A mass of 200.0 grams is hung from a vertical spring with constant k = 30.0 N/m. The mass is pulled downward 10.0 cm from its equilibrium position and released.

a) 2.0 s
b) 1.0 s
c) 0.5 s
d) 0.2 s

Answer 1

To find the force constant of a spring, the force exerted by the spring as described by Hooke's Law is compared to the downward force due to the mass's weight at different extensions. The unloaded length is determined by subtracting the extensions from the measured lengths with hanging masses. Both calculations must align to confirm the spring's consistent force constant and true unloaded length.

Step-by-step explanation:

To determine the force constant (k) of a spring, we use Hooke's Law, which states that the force (F) exerted by a spring is directly proportional to the displacement (x) from its equilibrium position: F = -kx. The negative sign indicates that the force exerted by the spring is in the opposite direction of the displacement. By rearranging Hooke's Law, we get k = -F/x. In this question, we're considering the force due to gravity, so F = mg, where m is the mass and g is the acceleration due to gravity (approximately 9.81 m/s²).

Calculating the force constant:

• When a 0.300-kg mass hangs, the spring's force equals the weight of the mass (F = mg = 0.300 kg × 9.81 m/s²).
• Using the displacements from the question and solving for the force constant for both masses, we should get equal force constants for each, which confirms the spring's force constant.

To find the unloaded length of the spring, we must know the total extension caused by each mass, subtract those extensions from the given lengths when the masses are attached, and ensure these calculations result in the same original unloaded length.