Hooke's Law
The equation for the force of a spring (Hooke's Law)
Fs = -kx
The equation for the potential energy of a spring
Us = (0.5)kx2
The equation for period of an object of mass m and spring constant k
T = 2π√(m/k)
In simple harmonic motion on a spring, at which position is the magnitude of the restoring force the greatest?
At maximum displacement (the amplitude)
The equation for the effective spring constant when two springs with constants k₁ and k₂ are connected in parallel.
keff = k1 + k2
A spring stretches 0.20 m when a 4 N force is applied to it. What is the spring constant of this spring?
k = 20 N/m
A spring with k = 200 N/m is compressed by 0.10 m. How much elastic potential energy is stored in it?
1 J
A 1.0 kg mass is attached to a spring with k = 100 N/m. What is the period of oscillation?
T ≈ 0.63 s (or 2π/10 s)
A mass oscillates on a spring and passes through the equilibrium position. At that instant, describe its acceleration and its speed.
Acceleration is zero; speed is at its maximum
Two identical springs, each with k = 100 N/m, are connected side by side (parallel) and support a hanging mass. What is the effective spring constant of the combination?
keff = 200 N/m
The spring constant of a spring is tripled, but the applied force stays the same. How does the stretch of the spring change?
1/3 of the original
A spring stores 8 J of energy when stretched 0.20 m. How much energy would it store if stretched 0.40 m instead?
32 J
The mass on a spring-mass system is quadrupled while the spring constant stays the same. By what factor does the period change?
The period doubles
On a force-vs.-displacement graph for a spring, what does the area under the curve represent?
Area under curve = potential energy stored in the spring
A spring with constant k is cut in half. How does the spring constant of each half compare to the original?
Each half has spring constant 2k
A spring is compressed 0.10 m by a 50 N force. How much force would be needed to compress the same spring by 0.25 m?
125 N
A 0.50 kg block on a frictionless surface is launched by a spring (k = 800 N/m) compressed 0.05 m. What is the block's speed after it leaves the spring?
2 m/s
A spring-mass system (k = 400 N/m, m = 1 kg) oscillates with amplitude 0.30 m. What is the maximum speed of the mass?
6 m/s
A student doubles the amplitude of a spring-mass oscillation. How do the period and the maximum speed each change?
Period is unchanged; maximum speed doubles
A spring-mass system is taken from Earth to the Moon, where gravity is about 1/6 as strong. The spring constant and mass are unchanged. How does the period of oscillation change?
The period does not change
A spring hangs vertically with natural length 0.40 m. After a 1.5 kg mass is attached and reaches equilibrium, the spring's length is 0.55 m. What is the spring constant? (Use g = 10 m/s²)
k = 100 N/m
A mass oscillates on a spring with amplitude A. At what displacement from equilibrium does the kinetic energy exactly equal the potential energy?
x = A/√2
A mass-spring system oscillates with period T. The spring is replaced with one that is 4 times stiffer, and the mass is replaced with one that is 9 times heavier. What is the new period in terms of T?
1.5T
At the moment a mass on a spring passes through a point where its displacement equals half the amplitude, what fraction of the total mechanical energy is potential energy?
1/4 of the total energy
Two springs connected in parallel have an effective spring constant of 12 N/m. The same two springs connected in series have an effective spring constant of 3 N/m. What are the individual spring constants k1 and k2?
k1 = k2 = 6 N/m