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.

A 3.2 kg plush toy attached to the free end of a horizontally oscillating spring oscillates once every 2.5 seconds. Given that the other end of the spring is fixed to a wall and friction is negligible, calculate the spring constant of the spring.

A physicist is experimenting on an elastic thread to make some observations. He fixes one end of the thread to the ceiling of his lab and attaches a block of 3.25 kg to the other end. This extends the length of the thread by 0.125 m when the block comes to a rest at the equilibrium. He then pushes the block vertically down from the equilibrium position, extending the thread further by 0.250 m. If he releases the block then, after what time does the block reach back up to the equilibrium position?

An engineer is calculating the characteristics of a new type of lightweight spring. He arranges it in a vertical orientation and keeps one of its ends fixed. He hangs a 3.2-kg mass from the other end. This causes the spring to extend by a length of 5.2 cm when it comes to equilibrium. Given that the mass is pulled down further by a distance of 3.3 cm from the equilibrium and then released, find the amplitude and frequency of the resulting oscillation.

Determine the vibration frequency of a 284-kg floating platform, which dips 3.6 cm deeper into a serene pond when a 70-kg individual steps onto it, and continues to oscillate for a time after the individual steps off.

A pendulum named "SpringBob" with a mass of 0.262 kg oscillates at the end of a slender, upright spring with a stiffness of 306 N/m and reaches a peak sway of 29.0 cm. When SpringBob sweeps through the balance point (y = 0) heading upward, the clock starts at t = 0. Which time function accurately characterizes SpringBob's rhythmic motion?

A 64-kg firefighter leaps from a height of 22.0 m into a rescue net, causing it to extend by 1.2 m. Assuming the net acts similarly to a basic spring, determine the extent of stretch if the firefighter rested in it.

When a vehicle travels at 80.0 km/h, vibrations are noticed due to an imbalanced wheel with a 15.0 kg mass and a 0.56 m diameter. Calculate the compression of the vehicle's springs when an additional load of 270 kg is evenly placed on the vehicle, considering that the vehicle has four identical springs.

Suppose a small basket mass m is suspended with two identical springs attached to the ceiling of a cozy cabin, each with a spring constant k. Determine the vertical oscillation frequency of the suspended basket.

What is the oscillation frequency of a uniform rod that is 1.26 m long and has a mass M? The rod is hinged at one end and connected to a spring with a spring constant k at the other end. It is initially held horizontally and then allowed to perform small vertical oscillations. To determine the frequency, apply the torque equation about the hinge point when the oscillation is small.

Consider a system called "SpringMass" consisting of a mass attached to a spring. This mass is pulled 8.9 cm from its equilibrium position and released from rest. The system oscillates with a period of 0.67 s. Write the equation that models the motion of this mass and determine its displacement after 1.9 s.

A group of content creators are dropping different objects on a trampoline to see what happens. They take a 53 kg dummy and drop it from the roof of a building 27.0 m above the ground onto the trampoline. As a result, the trampoline stretches 2.1 m from its equilibrium position. Modeling the trampoline as a spring, determine how much it would stretch from its equilibrium position if the dummy were dropped from 44 m instead.

At what time will a 0.262-kg bob swing on a slender vertical spring (spring constant 306 N/m) achieve its longest and shortest length if it starts oscillating with an amplitude of 29.0 cm and crosses the equilibrium point (y = 0) upwards at t = 0?

Determine the maximum acceleration of a 1.70-kg ball and the point at which it first occurs if it is set into motion from a lightweight spring suspended vertically, initially compressed by 17 cm from the neutral position (where y = 0) and completing each oscillation in 0.46 seconds. Additionally, at what point will this maximum acceleration first be achieved?

A rubber ball with a mass of 8.0×10¹ grams bounces on a trampoline, oscillating up and down with a frequency of 4.0 Hz. If a metal ball with a mass of 0.50 kg, but similar in size and shape to the rubber ball, is dropped onto the trampoline, what would be the expected frequency of its oscillations? Assume simple harmonic motion.