Simple Harmonic Motion of Vertical Springs Practice Problems

The following experimental setup was used by a science teacher to show simple harmonic motion in the vertical direction. The bottom end of an ideal spring is fixed to the top of a rigid surface. The upper end of the spring is covered with an adhesive. The spring has a natural length of l₀. A small sphere is dropped from rest 5.0 cm above the upper end of the spring and adheres to the spring. The system formed by the spring and the sphere oscillates with a maximum displacement of 8.0 cm. Calculate the angular frequency of the sphere.

A 250 g cube is attached to a massless, ideal spring suspended vertically from a horizontal table. At the equilibrium point, the spring is extended by 5.0 cm with respect to its natural length. The spring is released after being stretched downward by 2.0 cm from its equilibrium point. Calculate the period of oscillation of the cube.

During an experiment, a 125 g object is connected to a light and an ideal spring suspended vertically from a rigid support. At the equilibrium point, the spring is extended by 8.0 cm with respect to its natural length. A student is asked to stretch the spring down 7.0 cm from the equilibrium point. Calculate the maximum speed the object could reach.

A baby hanging toy is made of an ideal spring and a singing bird toy of mass 200 g. The spring attached to a horizontal board is suspended vertically. When the bird toy is attached, the spring is elongated by 4.0 cm from its rest length. The baby brings the bird toy down, releases it, and watches it oscillate. Calculate the frequency of the toy bird's motion.

A mass M is attached to the lower end of a light spring with a spring constant k suspended from a clamp. The system is made to oscillate. A motion detector measures the amplitude and time needed for the mass to complete 5 oscillations. This procedure is repeated with different masses, and the results are shown in the table. Calculate the spring constant using a suitable graph representation.

Consider an ideal spring with a spring constant of 15 N/m that is suspended from a rigid support. A load of 350 g is attached to the spring's free end. At time t, a motion detector shows that the load is located 15 cm under the equilibrium position and is moving straight up at 0.5 m/s. Calculate the frequency of the moving load.

A student used a vertical spring-mass system to investigate simple harmonic motion. A load of 250 g is attached to a vertical spring with a spring constant of 25.0 N/m. The student observes that the load has a repetitive movement back and forth through the equilibrium position. At time t = 0 s, the load is 15.0 cm below the equilibrium position and moving straight up at a speed of 0.75 m/s. Calculate the load's position with respect to the equilibrium position when it reaches a speed of 0.25 m/s.

A group of researchers announced the development of a new rubber-like material with surprising properties. A bungee jumper receives a 2.0 m long rope with a 1.0 mm radius made of this rubber material. He connects a 10 kg mass to the end of the rope and suspends it vertically by attaching the other end to a tall structure. From its equilibrium position, he takes the mass down 10 cm and lets it go. Using a motion detector, he counts 15 oscillations in 4.5 s. Calculate the rubber Young's modulus.

The suspension of a vehicle is an assembly of coil springs that keep the vehicle from bouncing too much when riding over an irregular surface. Consider a motorcycle of mass 300 kg with one coil spring on each wheel, and the motorcycle's mass is evenly distributed over the two coils. The two coils are identical and have a spring constant of 5.5 × 10⁵ N/m each. The motorbike transports two 65 kg riders. Calculate the frequency of the oscillations of the motorcycle. Consider the motorcycle in simple harmonic motion.