Intro to Simple Harmonic Motion (Horizontal Springs): Videos & Practice Problems

An object of mass 1.20 kg is attached to an ideal spring. The object's velocity as a function of time follows the relation v_x(t) = -(6.20 cm/s) sin[(7.53 rad/s)t - π/4)]. Calculate the period of the motion and the spring's force constant.

A  frictionless trolley of mass 14.2 kg attached to an ideal spring has a velocity function v_x(t) = -(8.50cm/s) sin[(5.20 rad/s)t + π/3)]. Determine the i) amplitude and ii) peak acceleration of the trolley.

A block of mass 0.650 kg attached at the end of an ideal horizontal spring undergoes SHM. When x = -0.130 m, its acceleration is 2.80 m/s². Determine the time taken by the block to complete one oscillation. 

SHM is used to determine the mass of objects. One procedure involves attaching a 6.80 kg container at the end of a spring. The empty container completes an oscillation in 0.80 s. When a block is placed in the container, the period of the oscillation becomes 1.90 s. Determine the mass of the block.

A wooden cube fixed on one end of a perfect spring lies on a horizontal smooth surface where it performs SHM. The motion has an amplitude A. The mass covers the distance +A to -A in 2.64s. The amplitude is doubled to A₁ = 2A, how much time will the cube take when traveling from x₁ = +A₁/4 to x₂ = -A₁/4?

A plastic disk is fixed to a perfect horizontal spring and placed on a horizontal smooth surface where it oscillates in SHM. The mass travels from -A to +A in 3.42 s. What is the time taken to travel from +A/2 to -A/2 when the amplitude is halved to A/2?

The head of a periodic hammer moves in simple harmonic motion parallel to the x-axis at a frequency of 8.0 Hz. When t = 1.58s, the position of the head is -2.3 cm while its velocity is +20.0 cm/s. Determine the acceleration of the head at t = 1.58s.

A  cube of mass 2.60 kg is fixed to one end of a perfect spring of k = 190 N/m and placed on a frictionless surface. When t = 0, the spring has its unstretched length while the cube has a velocity of 5.40 m/s in the negative direction. Determine the amplitude and phase angle. Express your answer in the form of a position-time function.

A crate executes SHM at the end of a spring with an amplitude of 168 cm and a period of 0.419 s. When t = 0, the crate has maximum displacement and is momentarily at rest. What is the time taken by the block to move from i) 168 cm to 62 cm and ii) 62 cm to 0 cm?

There are various methods of determining the spring force constant of a spring. One method involves measuring the mass and oscillation times for the mass. In one setup, you use a mass of 0.820 kg and measure the time it takes the mass to move from a point -A to the next instance it comes back to this same point to be 3.2 s. What is the force constant for the spring?

A spring of unknown force constant has a 3.20 kg block fixed to one of its ends. The block position as a function of time is shown in the graph below. Determine the amplitude of the oscillation and the spring's force constant.

A 1.6 kg block is fixed to one end of a spring. The motion of the block is monitored and the position as a function of time is graphed below. Determine the period and frequency of the oscillation.

Bees flap their wings at a rate of 13,800 beats/minute. Determine the period (T), the frequency (f), and the angular frequency (ω) for the flapping of a bee's wings.

A sewing needle performs SHM with an amplitude of 2.3 cm and a frequency of 20 Hz. Determine the time taken to move from y = 0 to y = 2.3 cm.

The variation of the displacement with time for a vibrating mass is shown in the graph below. Determine the frequency and angular frequency for the vibration.

The graph below shows displacement as a function of time for a vibrating mass. Determine the period and amplitude of the vibration.

Vibrating objects emit sound that matches their vibrating frequency. In music, this frequency is categorized into notes. A keyboardist is playing C# (C-sharp) note with a frequency of 554 Hz. The sound is emitted by the diaphragm in a speaker that vibrates at this frequency. Determine the time taken by the diaphragm to complete one cycle and its angular frequency.

An object of mass 8.1 kg is attached to a spring with a spring constant of 20 N/m. The object is displaced from its equilibrium position and then released from rest at t = 0s. Determine the position of the object at t = 0.45 s if the object oscillates with a maximum speed of 30 cm/s. Consider the period of oscillation is 4.0 s.

A metallic block (m = 650 g) has a cavity with a spongy lining that holds balls shot to the block. The block is fixed to a horizontal spring. A 75 g ball is shot to the block in a direction parallel to the spring on the face fixed to the spring, stretching the spring. After that, the block is observed to oscillate at a frequency of 0.8 Hz. If the block has an amplitude of 30 cm, determine the speed of the ball when it enters the cavity.

Determine the phase constant for an object of mass 0.50 kg attached to a spring. The object is oscillating with a period of 2.0 s. Initially, the object is 3.0 cm to the left of its equilibrium position and it has a velocity of 20.0 cm/s in the positive direction of the x-axis.

A block with a mass of 0.700 kg is attached to a spring and is oscillating. Calculate the spring constant if the block completes 16 oscillations in 30.0 s.

A 300 g mass is attached to a spring that experiences oscillations with a period of 1.5 s. Determine the period of oscillations if the spring constant is made 4 times its original value.

A 1.6 kg piece of wood lying stationary on a frictionless bench is mounted to a horizontal spring. The other end of the spring is fixed on the wall. A 25 g stone thrown parallel to the spring from a slingshot enters a hollow in the piece of wood and gets stuck. If the maximum displacement in the oscillatory motion of the piece of wood is 850 mm, and the spring constant is 3200 N/m, find the stone's speed when it hits the block.

An ideal spring with a mass of 2.0 kg attached to one of its ends rests on a smooth horizontal surface. The other end is fixed to a support. The spring constant is k. When the spring is stretched by 5.0 cm, the mass oscillates with a period of 3.0 s. Calculate the oscillation period if the spring is stretched by 7.5 cm.

An object of mass 300 g is attached to a horizontal spring. It is experiencing oscillations with a frequency of 3.0 Hz. Determine the maximum speed of oscillations if the object's initial position is x = 10 cm and the velocity is v_x = 20 cm/s.

Determine the total energy of a 500 g mass attached to a horizontal spring. The mass passes through its equilibrium position with a velocity of 25 cm/s. Consider the mass oscillates at a frequency of 3.5 Hz.

A spring is connected to a block of mass 350 g. The block is oscillating horizontally at 4.0 Hz. If the initial position of the block is at x = 8.0 cm with v_x = 25 cm/s, what is the position of the block at t = 0.2 s?

A 100 g mass oscillates with a displacement given by x(t) = (3.0 cm)sin(5t + π/6), where t is in seconds. Find the initial position and velocity of the mass.

The equation x(t) = (5.0 cm)sin(8t + π/3), where t is in seconds, represents the displacement of a 70 g mass. What will be the velocity of the mass at t = 0.35 s if it is oscillating?

Consider a spring of spring constant 12 N/m connected to a 2.5 kg block. The block is initially at rest, but then a force of 15 N is applied to it, imparting a speed of 25 cm/s to the block. Determine the block's speed when it reaches a displacement of x = 0.03 m.

A spring-mass system lies on a horizontal disk initially at rest. The spring's free end is attached to the disk axle. The mass undergoes periodic motion with a period of 0.65 s. When the disk is spun at a certain frequency (mass not oscillating), the spring extends, and its length increases by 25%. Calculate the disk's rotation frequency in rotations per minute (rpm).

Different techniques have been developed to measure the mass of a single biological molecule using the simple harmonic vibration of a rigid nanobeam that extends horizontally and is supported at only one end. The nanobeam vibrates with its fundamental frequency f given by f=(1/2π)√(k/M*). k represents the beam "spring constant," and M* is the effective mass. For a rectangular nanobeam, M* is equal to 0.24 M, where M is the mass of the nanobeam. Consider a rectangular nanobeam of area 5000 × 10^-9 m² and thickness 80 nm. The beam's density is 1800 kg/m³. Using an ultrasensitive sensor, the frequency of the beam before attaching the biological molecule was measured to be 8.5 × 10⁶ Hz. Connecting a single biological molecule to the free end of the nanobeam causes a 100 Hz reduction in the value of the initial frequency. Calculate the mass of the biological molecule.

A box having a mass of 250 g is undergoing oscillations on a frictionless surface. The box is connected to a spring with a spring constant of 3 N/m.The speed of the box is 25 cm/s when x = -8.0 cm. Determine the displacement of the box when the acceleration is maximum.

An engineer is testing the suspension system of a new car model. He attaches one end of a large spring to the car's bumper and the other end to a fixed point. The car is pulled away from the fixed point and released. The figure below shows the graph of spring length vs time. Determine the speed of the car when the length of the spring is 1.5 m.

A butterfly of mass 0.33g alights on a stationary horizontal string. This causes the string to oscillate at 9.0 Hz. If the mass of the butterfly was 0.66 g instead, at what frequency would the string oscillate?

A block attached to the free end of a horizontal spring which is oscillating with a frequency of 221 Hz. The maximum distance reached by the block on either side of the equilibrium position from the equilibrium is 2.20 mm. Find the maximum speed attained by the block.

An opera singer is trying to shatter a wine glass with her voice. She holds the glass close to her mouth and starts singing. As a result, the tip of the glass starts oscillating at a frequency of 523 Hz. If the rim of the glass moves to a maximum distance of 1.10 mm from the equilibrium at that frequency, determine its maximum acceleration.

A spring-loaded boxing glove oscillates horizontally with a maximum displacement of 0.24 m from the equilibrium. Given that the glove oscillates 4.5 times per second, and the mass of the glove is 0.28 kg, calculate its speed when it is 0.18 m away from the equilibrium.

Consider a 310 g block on a horizontal surface, linked to a massless elastic rod characterized by a spring constant of 150 N/m. After being pushed to a speed of 2.5 m/s, determine the period (T) and frequency (f) of the subsequent oscillatory motion. Ignore friction.

Imagine a 430 g block resting on a smooth horizontal surface, connected to a massless spring-like cushion with a spring constant of 180 N/m. Suddenly, the block is given a push, accelerating it to an initial speed of 3.3 m/s. Calculate the amplitude (A) of the subsequent oscillatory motion.

A child is playing with a clown toy of mass 0.55 kg attached to the free end of a spring. The other end of the spring is fixed. The clown toy is oscillating 3.5 times per second with an amplitude of 0.25 m. Given that at the beginning, the clown toy was at a maximum distance from the equilibrium position, determine the equation that models the motion of the clown toy.

What is the oscillation frequency of a 216 g cart positioned on a frictionless rail attached by identical springs to both ends of the rail with each spring constant of 126 N/m, if there's no damping?

An aluminum mass m is placed on a frictionless horizontal surface and connected to two copper wire springs arranged in series. The first spring, made from a thicker gauge of copper wire, has a different spring constant from the second, which is made from a thinner gauge. Determine the period T of horizontal oscillation for the aluminum mass.

A block of mass M is attached to the free end of a horizontal spring. The other end of the spring is fixed to a wall. At t = 0, a child kicks the block with an impulse J. Given that the spring constant is C, determine the expression describing the resulting motion of the block in terms of J, M, C, and t. Assume that there is no friction involved.

A group of students are experimenting with two different pendulums (pendulum 1 and pendulum 2) placed side by side. They set both of the pendulums in swing by pushing them from the equilibrium in the same direction at the same time. They find that the equations giving the distance of the bobs from the equilibrium position are x₁ = (1.5 m) sin (5.0 t) and x₂ = (2.5 m) sin (8.0 t). Determine the next two consecutive times after the pendulums begin swinging (apart from the first time, corresponding to t = 0) at which the bobs pass the equilibrium positions simultaneously.

A cart of mass 1.5 kg is initially at rest on a smooth horizontal surface. One end of the cart is attached to a horizontal spring with a spring constant of 230N/m. Suddenly, it is kicked by a person, giving it an initial speed of 3.0 m/s. What is the maximum acceleration experienced by the cart?

A 1.2 kg object is initially at rest on a frictionless horizontal surface. The object is attached to an air cushion with an equivalent spring constant of 190 N/m and is punched by a fist, giving it an initial speed of 4.5 m/s. Calculate the position of the object as a function of time.

Imagine that you are testing out a prototype spring system, and that you have recorded the displacement versus time graph for a small mass that is attached to the end of the spring. The figure below shows the graph that is produced from this motion. Note that at t=0, the displacement will be x=0.79 cm.

(i) If the mass of the suspended component is 8.2 kg, determine what the spring constant, k, of your prototype spring system will be.

(ii) Express what the equation for displacement x as a function of time will be.

Consider the simple harmonic motion of a mass connected to a spring, as depicted in the figure below. Express the equation representing this motion in either sine or cosine form.

In a nitrogen molecule (N₂), consider a situation where one nitrogen atom is fixed in place, and the other nitrogen atom is bonded to it with a spring-like chemical bond. When exposed to infrared light, tthe nitrogen atom that is not fixed vibrates with a frequency of 2.2 x 10¹³ Hz. If the vibrating nitrogen atom is replaced with a carbon atom, while the spring constant of the bond remains the same, determine the new frequency of vibration for this nitrogen-carbon molecule.

Carbon disulfide (CS2) is a linear triatomic molecule. The sulfur atoms oscillate symmetrically in and out while the central carbon atom remains stationary, in a similar fashion to the vibrations in carbon dioxide. This vibrational mode is observed with a frequency of $2.5\times 10^{13}\ Hz$. Calculate the spring constant of the C-S bond assuming each sulfur atom behaves as a simple harmonic oscillator.

A steel block is attached to a helical spring on a frictionless air track. The block is initially set to a position -A units from its equilibrium position and released from rest. Which graph illustrates the positions of the block at given time: 0, 1/4 T, 1/2 T, 3/4 T, T, and 5/4 T, where T is the natural period of oscillation?

The displacement of an oscillating object is given by the equation x(t) = B sin(ωt + φ). If at time t = 0, the object's displacement is B, find the phase constant φ.

In a simple harmonic motion, at time t = 0, the displacement of the oscillating object is at its maximum amplitude A. Determine the phase constant (ϕ) in this scenario.

An object is undergoing simple harmonic motion. At time t = 0, the object is at position x = -3/4 A. Determine the phase constant φ.