Simple Harmonic Motion of Pendulums Practice Problems

The equation of SHM motion of a simple pendulum is derived using assumptions that make it valid for small displacements. During an experiment, you suspend a spherical ball using a 1.20 m long thread and displace the pendulum by 28°. Calculate the period of the pendulum assuming small displacement and using the equation . i) State the value that is more accurate, ii) Determine the percentage error in the other value by comparing it with the value you consider accurate. 

The motion of a simple pendulum approaches SHM for small displacements. Suppose you displace a sphere suspended from a 1.4 m long thread through 25.0°, i) what is its period assuming a small displacement? ii) What is the period using the given three terms of equation ?

A grandfather clock with a 1.00 m long pendulum hangs on a clock tower. The mass of the bob on the pendulum is 0.500 kg. An unlucky driver hits the tower vibrating it. If the pendulum was initially at rest in the equilibrium position, how many cycles per second will the pendulum make?

A plumb bob displaced by a small angle from the equilibrium position has a period of 1.23 s on the earth. What is the period of this simple pendulum when taken to the surface of planet K where g is 13.2 m/s²?

You construct a 0.84 m long simple pendulum using a light string then displace it by 4.30° on one side before releasing it to oscillate. Next, you double the displacement angle to 8.60° on the same side. What is the difference between the two time periods for the two angles used?

A simple pendulum has a length of 1.35 m. The pendulum is displaced to one side by 5.60° and allowed to oscillate. Determine the time taken from the time it is launched to the time its acceleration is zero. 

A decorative pendulum on a wall clock is made of a 250 grams gold coated spherical mass hanging on a 35 cm long lace. The homeowner starts the pendulum at 4:00 pm by giving it a displacement of 3.0 cm on one side. The pendulum is nearly perfect, with a damping constant of 2.5 * 10^-5 kg/s. Find the amplitude and number of cycles the pendulum makes by 11:00 pm.

A bob of mass 125 g is suspended from a 105 cm long massless thread. The ball is given an initial displacement of 8 degrees. Calculate the number of times the bob goes through the equilibrium position in 10.0 s.

A bob suspended from a 1.2 m long string oscillates like a pendulum. The speed of the pendulum at the equilibrium point is 1.50 m/s. Calculate the displacement angle (θ_max) of the bob in degrees.

A self-navigating space vehicle moves a pendulum between planetary objects. A pendulum has a period of 1.2 s on Jupiter (g = 24.8 m/s²). The pendulum is moved to a different planetary object where its period is measured to be 3.2 s. Calculate the gravitational acceleration on the planetary object.  

An L-meter-long regular wooden bar of mass M and uniform composition is pivoted using a nail at a point (0.3L) as measured from one end of the bar. For the bar oscillating at small angles, derive an expression for the frequency f of the oscillation.

A decoration consists of a 22 cm long bar of mass 270 g. One end is used like a pivot while the opposite end is fitted with a spherical bob, which can be considered a point mass, with a mass of 48 g. If the bar-bob combination oscillates like a pendulum, find the period of the motion.

A regular piece of wood with a length L is pivoted about a position x measured from the center of mass of the piece of wood. Calculate the value of x that gives the lowest oscillatory period. Express x in terms of L.

Moment of inertia is an integral part of designing objects and systems that undergo rotations. The length of a 12 kg bar in a crane model is 1.3 m. The center of mass of the bar is located 25 cm from the rotational axis. The bar oscillates about the rotational axis like a pendulum at a frequency of 2.0 Hz. Find the moment of inertia of the bar about the rotational axis.

Imagine a spherical ball named "OrbMover" suspended from a ceiling and initially hanging at a rest angle of 9.5° from the vertical. Determine the new angle that OrbMover must reach if its total potential and kinetic energy combined are to be doubled.

A pendulum known as "TimeSwing" begins its motion from a stationary state at a 10° deviation from the vertical axis. It swings back and forth with a regular frequency of 2.8 Hz. Disregarding any resistance, determine TimeSwing's angular position in radians of the pendulum at t=0.28 s?

A pendulum named "SwingMotion" executes oscillations with a maximum angular displacement of 12.0° from the equilibrium position. What fraction of its period is spent between +6.0° and -6.0°? Presume the motion follows simple harmonic principles.

"GraviSwing" is a simple pendulum swinging with an initial frequency ƒ. Determine the new frequency of GraviSwing's oscillations when it experiences an upward acceleration of 0.6 g.

Imagine a spherical ball with a mass of 296 g attached to a 0.64 m long string hanging from the ceiling. The ball is released at an angle of 14° to the vertical. Assuming the motion is simple harmonic, calculate the frequency at which the ball oscillates.

Imagine a ball acting as a pendulum suspended from the ceiling of a cabinet with a string of length 52 cm. Calculate the period of the ball's oscillations when the cabinet undergoes freefall.

Imagine a metal ball suspended by a string, swinging back and forth at a certain frequency (f). Now, if we were to place this entire setup in an elevator descending with an acceleration of 0.28g, what would be the new frequency (f') of the ball's oscillations?

In a clock mechanism, a 0.96 cm radius pendulum wheel attached to a coil spring vibrates at 3.20 Hz. If 2.1 × 10⁻⁵ N.m torque causes a 60° rotation, what is the wheel's mass?

A 2.30 kg acrylic disk with 30.0 cm radius is suspended by a metal hook through a hole 5.00 cm from its edge, allowing it to swing as a pendulum. What is the period of its small oscillations?

An inventor makes a device consisting of a speaker hanging freely from the ceiling by a string. The speaker if set swinging, plays a drumbeat each time it reaches the maximum distance from the equilibrium. Given that the speaker plays 120 bpm (beats per minute) after it is set swinging, what is the length of the string?

A student places a ruler at the edge of a table. He holds one end of the ruler inside the table and makes the other end stick outside. He then slaps the end of the ruler sticking out the table at a speed of 32 km/h. This makes the ruler oscillate in SHM. Given that the part of the ruler sticking out the table is 15 cm in length, the period oscillation is 0.78 s and the amplitude is 7 cm, determine the maximum velocity and acceleration of the ruler's oscillating end. Express the maximum acceleration as a percentage of the gravitational acceleration g.

A physics professor is on a tour. He is to deliver lectures in various places. For the lectures, he is carrying along a pendulum with him. He wants the pendulum to have a period of exactly 1.000 s. In Florida, the professor measures the value of 'g' to be 9.798 m/s². And in Anchorage, it turns out to be 9.819 m/s². On the other hand, the value of 'g' on Mars is 3.721 m/s². Determine the length of the professor's pendulum in Florida and the length adjustment (in millimeters) he makes going from Florida to Anchorage to the pendulum's length. Also, find the length of the pendulum on Mars if the period is to be the same.

A traditional grandfather clock employs a compound pendulum for timekeeping, which consists of two interconnected metal rods that each swing about a fixed upper pivot point. Assume that both rods are equally long, each with a length of 0.75 meters. The mass of the upper rod is 1.5 kg and the lower rod carries a mass of 1.0 kg. Compute the period of oscillation for this particular pendulum system.

A museum exhibit features a large, decorative pendulum swinging back and forth. The pendulum consists of a heavy bob of mass 𝑚 at the end of a long, light rod of length 𝑙. Taking into account the gravitational acceleration 𝑔, derive the equation for the angular displacement 𝜃(𝑡) of the pendulum, given that the maximum angular displacement is small. Assume the pendulum undergoes simple harmonic motion for small displacements. Use the relationships $\tau$ = 𝐼𝛼.

A fork-shaped metallic plate oscillates between the North and south poles of a magnet. Its amplitude is reduced to 42.8% of the initial amplitude in 8.60 s. Calculate the time constant.

Gravitational force can be used as a restoring force for a mass m_o oscillating in a hollow drilled through a spherical mass's center (a diameter). For a sphere of mass M_s, and radius R_s, it is proven that an oscillating mass with position x where r ≤ x ≤ R experiences a net gravitational force from the spherical mass enclosed by r where r ≤ x, while the mass of spherical shell at x > r does not contribute to the net gravitational force of the oscillating mass. For a sphere with even density, derive an expression for the gravitational force on the oscillating mass. Express the result using M_s, R_s, x, m_o, and any desired constant(s).