Energy in Simple Harmonic Motion Practice Problems

A baby of mass 12 kg is sitting in a baby bouncer. The baby bouncer is suspended in a room with the help of a light elastic cord. The unstretched length of the elastic cord is 2 m. When the baby bounces, she executes a simple vertical motion with an amplitude of 8 cm. The cord has its natural unstretched length at the highest point of motion. The elastic potential energy of the unstretched cord is assumed to be zero, and the lowest point reached by the baby as the reference level. Calculate the i) kinetic energy,  ii) elastic potential energy, and iii) gravitational potential energy at the baby's highest point.

A spring-mass system is used to simulate the motion of the human hamstring. The hamstring executes a simple harmonic with an amplitude of 15 cm and an angular frequency of 6.28 rad/s. Calculate I) i) the maximum magnitude of the acceleration and ii) the speed at the equilibrium position; II) i) the acceleration and ii) speed at the position x=-7.5 cm; III) the time needed to go directly from x=0 cm to x=+7.5 cm IV) Which of the quantities requested previously can be determined using the conservation of mechanical energy?

An object of mass m executes a simple harmonic motion when attached to a spring with a spring constant k. The amplitude of the simple harmonic motion is A. Find the position of the object from the equilibrium position if the kinetic energy is double the potential energy. Express your answer in terms of A.

A 225 g object placed on a frictionless horizontal table and attached to a spring executes a simple harmonic motion. The spring constant is 125 N/m. The maximum measured displacement from the position of equilibrium is 8 cm. What is the total mechanical energy of the spring object system?

A simple harmonic oscillator is made up of a 0.25 kg object attached to a spring with a force constant of 100 N/m. The object's maximum position, measured from the equilibrium position, is 5 cm. Calculate the object's speed when its position is -2 cm.

A particle executes simple harmonic motion with an amplitude of 25 cm. As the particle passes through the equilibrium point, its speed is 5 cm/s. Calculate the maximum acceleration of the particle.

The figure below shows the position-time graph of a particle oscillating along a horizontal plane. Find the maximum acceleration of the particle.

The position-time graph of a particle attached to a horizontal spring is shown in the image. What is the object's maximum speed?

Consider a proton that is confined within a one-dimensional harmonic oscillator potential. The oscillator has a spring constant of 1.5 N/m. Determine the energy values corresponding to the first three levels of the proton's energy within this system.

A mechanical oscillator consists of a block of mass m and a spring of force constant k. Placed on a smooth horizontal table, the block undergoes simple harmonic motion with an amplitude 30 cm and frequency 0.5 Hz. At the position x = 20 cm, what are i) the speed (v) and ii) the acceleration of the object (a_x)?

Consider that the oxygen atom in a NO molecule undergoes a simple harmonic motion with a frequency f and an amplitude A. At the equilibrium position, the elastic potential energy of the oxygen atom is set to zero. Find a) i) the displacement and ii) the velocity of the oxygen atom when the kinetic energy and elastic potential energy are equal. b) i) How many times does this happen during a given period? ii) and what is the time interval separating two consecutive occurrences? c) Calculate the percentage of mechanical energy that is kinetic energy when the oxygen atom's displacement is one-third of its maximum displacement. Note that at t = 0, x = A.

Express the kinetic and potential energy in terms of the total energy for a horizontal spring in simple harmonic motion. Consider the displacement of the mass is C/5, where 'C' is the amplitude.

Suppose you have two masses, one of 300 g and the other of 400 g, connected to each other by a spring with a spring constant of 3.0 N/m. The masses are attached to walls via springs having a spring constant of 20 N/m. Initially, the 300 g mass is displaced 10 cm to the right and released, while the 400 g mass is at rest at the equilibrium position. Assuming no friction, how long will it take for the 300 g mass to stop moving and for the 400 g mass to start moving?

A block of mass 450 g is connected to a vertical wall by a spring having a spring constant of 20 N/m. Initially, the 450 g mass is at rest at the equilibrium position while another mass of 350 g is pushed towards the 450 g block at a velocity of 90 cm/s causing the two masses to collide and stick together. Assuming no friction, determine the amplitude and period of the resulting oscillations.

A 3.25-kg block is attached to the free end of a horizontally oscillating spring. The other end of the spring is attached to a wall. The position x of the block from the equilibrium position is given by the equation x = (0.540)cos(3.20t), where x is measured in meters and t is in seconds. Calculate the total energy of the system, assuming there is no energy dissipation.

A metal ball of 2.43 kg is attached to the free end of an elastic spring, the other end of which is fixed. The ball is pulled from the equilibrium and released which causes it to oscillate. The displacement of the ball from the equilibrium is given by x = 0.430 cos(5.30 t), where the quantities are measured in the SI units. Determine the kinetic and potential energy of the metal ball corresponding to the displacement x = 0.215 m.

Two horizontally oscillating springs A and B, have two blocks with the same mass attached to their free ends and oscillate with the same frequency. The other ends of the springs are fixed. The total energy of the attached block of spring A is 4.0 times that of the attached block of spring B. Determine the ratio of their amplitudes.

A guitar string is in simple harmonic motion. Calculate the fraction of the total energy that is kinetic and potential when the displacement of a point on the string is one-fourth of its amplitude from the equilibrium position.

A block is attached at one end of a horizontally oriented elastic rod. The other end of the rod is secured. Applying a compressive force on the block does a work of 4.3 J on the system. This causes the rod to shrink by 0.11 m. If the force is withdrawn, the block experiences an acceleration of 13 m/s². Calculate the mass of the block. (Assume friction is negligible.)

A physicist is investigating the elastic properties of a material. He takes a horizontal rod made of that material and attaches a block to one of its ends while keeping the other end fixed. He compresses the rod by 0.17 m by applying a force on the block that does a work of 4.4 J on the system. As soon as he releases the block at rest, it experiences an acceleration of 17 m/s². Assuming there is no friction, determine the spring constant of the rod.

A metal ball of mass 1.5 kg is fixed to one end of an elastic metal rod. It is carrying out simple harmonic motion. The other end of the rod is fixed to a wall. The spring constant of the metal rod is k = 440 N/m. Given that the speed of the metal ball at a distance of 0.030 m from the equilibrium is 0.80 m/s, determine the maximum speed of the metal ball in its trajectory.

Imagine a box weighing 45 kg resting on a calm lake. Adding a block weighing 13 kg to the box causes it to sink an extra 5.0 cm into the water. After removing the block, the box starts to oscillate gently. Calculate the total energy of this oscillation, assuming no damping effects and ignoring the gravitational potential energy.

At what maximum amplitude can the bottom end of a diving board oscillate with simple harmonic motion at a frequency of 3.2 Hz in such a way that a coin placed on the bottom end of the diving board remains secure and does not slip off and fall into the pool during the oscillation as seen in the figure below?

A student is constructing a device for an exhibition at a science expo. It consists of an inclined plane at the foot of which a spring is attached. For testing the device, the student takes a billiard ball of 170 g and puts it at the free end of the spring. She compresses the spring by 9.0 cm and releases. Given that the incline makes an angle of 33° with the horizontal, the radius of the billiard ball is R = 6.0 cm and the speed at which the billiard ball is launched by the spring is 4.0 m/s, determining the spring constant of the spring. [Assume that the billiard ball rolls without slipping.]

Engineers are measuring the properties of suspension springs that are made in a factory. They fix one end of a sample spring to a wall and attach a block of 80 g to the other end. Then they make it oscillate with an amplitude of 3.0 cm. They find that the spring constant of the sample spring is 80 N/m. A computer generates its potential energy versus position graph that looks like the figure below. From the graph, find the potential energy of the spring corresponding to x = 2.0 cm. Assume that that friction is negligible and the spring is massless.

A student is experimenting with a spring. He arranges it in a horizontal orientation, attaches a rubber ball of 30 g to one of its ends, and fixes the other end of the spring to a wall. He then compresses the spring by 3.0 cm and releases it so that it oscillates with the same amplitude. Afterward, he draws its energy versus distance from the equilibrium curve given in the figure below, assuming there is no friction and the mass of the spring is negligible. If the spring constant of the spring is 40 N/m, find the kinetic energy of the spring corresponding to x = 2.0 cm from the graph.

A physics teacher is demonstrating a spring in her lab to her students. The spring, having a spring constant of 40 N/m, is in a horizontal orientation, and one of its ends is fixed. A small metal ball of mass 90 g is attached to its free end. The teacher makes the metal ball oscillate by compressing the spring by 3.0 cm first and then letting go. The spring, as a result, oscillates back and forth with the same amplitude. Afterward, the teacher draws a graph on her board corresponding to the motion of the spring. If she assumes that friction and the mass of the spring are negligible, determine the speed of the ball at the distance from the equilibrium x = 2.5 cm, from her graph given below.

A floating cylinder undergoes SHM with an amplitude of 0.6 cm. The angular frequency of the motion is 4.45 rad/s, while the phase constant is π/3 rad. Draw about 2 cycles of the cylinder's velocity-time graph starting at t = 0s. Assume the motion follows x(t) = Acos(ωt +  ϕ).

The graph below is a position-time representation of the SHM of a floating block dipping and rising in water. Find its peak speed.

A wooden block floating in water performs SHM using buoyant force. It oscillates with a period of  1.70 s and a maximum displacement of 4.20 cm from the equilibrium position. Calculate the time the block takes to move from x = 0 to x = 3.5 cm.

Use the position-time graph below to determine the phase constant of the motion. Assume the displacement is described by x(t) = A cos(ωt + ϕ).

Determine i) maximum displacement ii) the number of oscillations per second, and iii) the phase constant of the object using the periodic motion graph below.