A simple pendulum has a mass of 0.250 kg and a length of 1.00 m . It is displaced through an ang

step 1 Now for this question, we have a simple pendulum. Given that it is displaced by a small angle of

A simple pendulum has a mass of 0.250 kg and a length of...

Key Concepts

Simple Pendulum

A simple pendulum is a classical physics system consisting of a mass (or bob) suspended from a fixed point by a weightless and inextensible string or rod, free to swing under the influence of gravity. This system is often used to study basic principles of dynamics and oscillatory motion, where its behavior is determined primarily by gravitational force and the geometry of the setup.

Simple Harmonic Motion (SHM) Approximation

Under the small angle approximation, the motion of a simple pendulum is modeled as simple harmonic motion, meaning that the restoring force is directly proportional to the displacement from equilibrium. This approximation simplifies analysis because it leads to linear differential equations, and it is valid when the angular displacement is sufficiently small, ensuring that the sine of the angle is approximately equal to the angle itself measured in radians.

Energy Conservation in Oscillatory Systems

Energy conservation principles are pivotal in analyzing oscillatory motion. For a pendulum, gravitational potential energy is converted into kinetic energy as it swings from its maximum displacement to the lowest point, and vice versa. This conservation of energy allows one to calculate important quantities like the maximum speed of the bob by equating potential energy at the highest point with kinetic energy at the lowest point without directly solving the equations of motion.

Angular Acceleration and Restoring Force

In the context of a pendulum undergoing simple harmonic motion, the angular acceleration is directly proportional to the negative of the angular displacement, characterized by the relation a = -(g/L)?. This relation arises because the restoring force, which is the component of the gravitational force acting along the arc of motion, is responsible for accelerating the pendulum back toward equilibrium. The maximum restoring force is therefore experienced at the point of maximum displacement, where the sine function is maximized within the small angle range.

Comparison of Analytical Approaches

Different analysis models can be applied when studying pendulum dynamics. The SHM model, which uses the small angle approximation, greatly simplifies calculations and yields linear relationships between variables such as acceleration and displacement. Alternative models from earlier studies may employ more exact trigonometric or energy-based approaches that do not rely on the small angle approximation, leading to slight differences in predicted values. Comparing these methods helps to understand the validity and limitations of approximations in physical modeling.

Transcript

00:01 Now for this question, we have a simple pendulum.

00:04 Given that it is displaced by a small angle of 15 degrees, and you want to use the simple harmonic motion approximation to find some of the properties of this pendulum.

00:21 So using the simple harmonic motion model, the amplitude of this will just be be the total distance, right, that is displaced, where it reaches the 15 degrees.

00:44 And we can approximate this to be an arc, right? so the amplitude will just be taking the radius, multiply by the angle in terms of radiance.

00:59 So the radius is the length of the pendulum, which is given as 1 meter.

01:06 And the angle, which is 15 degrees, in terms of radiance, we'll have to convert it into radiance by defining by 180 degrees, multiply by pi.

01:19 So using this amplitude, you can find the maximum velocity by multiplying this with the angular frequency.

01:34 Now what is the angular frequency for a pendulum? this is just square root of g over l, right where g is the gravitational acceleration, l is the length of the pendulum, just give us about 3 .13, medients per second, this is the angular frequency.

02:04 So just substitute here, this 0 .262 multiply by 3 .13.

02:24 This should give us 0 .820 meters per second.

02:34 So this is the maximum velocity.

02:40 To find the maximum acceleration for a simple harmonic motion, it's just a omega square.

02:51 So instead of just a omega, multiply omega an extra time.

02:56 So this 262, 3 .13 squared.

03:00 Should get 2 .57 thus per second per second.

03:27 Now to find its angular acceleration let's be know that the maximum angular acceleration is also the maximum linear acceleration you tend to convert from this linear acceleration to angular acceleration just have to divide this by the length of our pendulum, which is just one meter.

04:13 This will give us 2 .57 medians per second per second.

04:24 Now to find the maximum force, now we know that force is equals to mass times acceleration.

04:33 So maximum force occurs when there is maximum acceleration.

04:38 And we are given that the mass of this blob is 0 .25 kg, so we multiply this with the acceleration, which is 2 .57.

04:48 Just again, just again, and we should get the force to be 0 .641 newtons.

05:01 Now, all of this is approximating the pendulum as a simple harmonic motion, but of course it's not an exact simple harmonic motion for the pendulum.

05:16 So it is not, it's a good approximation but not the exact answer.

05:21 So we want to use a different method of applying to look at this problem which will give us a more accurate answer, right? more exact answer.

05:32 And the way that we're going to do this is by looking at energy, conservation.

05:44 So we know that it is actually lifted up to an angle of 15 degrees and released.

05:50 So we know that there is some height that it gains and this will tell us what is the gravitational potential energy that was gained.

06:04 And all this gravitational potential energy is converted into kinetic energy once it moves to the center over here and that is when we will have the maximum kinetic energy...

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