A thin rod of mass 0.630 kg and length 1.24 m is at rest, hanging vertically from a strong, fixe

A thin rod of mass 0.630 kg and length 1.24 m is at rest,...

A thin rod of mass 0.630 $\mathrm{kg}$ and length 1.24 $\mathrm{m}$ is at rest, hanging vertically from a strong, fixed hinge at its top end. Suddenly, a horizontal impulsive force 14.7 $\mathrm{i} \mathrm{N}$ is applied to it. (a) Suppose the force acts at the bottom end of the rod. Find the acceleration of its center of mass and (b) the horizontal force the hinge exerts. (c) Suppose the force acts at the midpoint of the rod. Find the acceleration of this point and (d) the horizontal hinge reaction force. (e) Where can the impulse be applied so that the hinge will exert no horizontal force? This point is called the center of percussion.

Key Concepts

Translational Motion

This concept describes how the center of mass of an object accelerates when a net force is applied. It is governed by Newton's second law, where the acceleration of the center of mass is directly proportional to the net external force and inversely proportional to the mass of the object. In dynamics problems, separating the translational motion from the rotational motion is key to analyzing how different forces affect different parts of the motion.

Rotational Dynamics

Rotational dynamics deals with the rotation of a body about an axis, which is influenced by torques produced by forces acting at a distance from the axis. It involves understanding how angular acceleration, moment of inertia, and torque interact. This analysis is crucial when forces do not act through the center of mass, producing a combination of linear and angular accelerations.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to angular acceleration, depending on how its mass is distributed relative to the axis of rotation. It is analogous to mass in linear motion. When a force is applied at different points along a rigid body, the distribution of mass relative to the pivot affects the resulting angular acceleration and energy distribution in rotational motion.

Impulse

Impulse is defined as the change in momentum resulting from a force applied over a short time interval. In problems involving sudden forces, or 'impulsive forces', impulse analysis helps determine the immediate effects on both linear and angular momentum. It is particularly useful for analyzing collisions or sudden impacts where time duration is negligible.

Center of Percussion

The center of percussion is the specific point on a swinging object at which an impulsive force can be applied without causing any reaction force at the pivot point. This concept explains how the interplay between translational and rotational dynamics can lead to a situation where the net effect at the hinge or pivot is zero. It has practical implications in designing objects like bats and swords to minimize shock transfer.

Transcript

00:01 In this question, we have this thin rod of mass 0 .63 kg of length, 1 .24 meters.

00:07 It's hanging vertically from a strong fixed hinge at its top, and then a horizontal force is applied of 14 .7 newtons to the right, it applied to it.

00:18 Okay, so there are five parts in this question.

00:21 In part a, if the force acts at the bottom of the rod, find an acceleration of the center of mass.

00:28 Okay, so to do this, we'll take a pivot at the hinge, and then we'll be using torque equals to i alpha, so it goes to f times the level arm, okay, in this case it's l.

00:54 So yeah, to find the acm in this case, we need to find alpha first.

01:00 So alpha will be f times l divided by i.

01:05 K would be fl divided by 1 3rd, ml square.

01:10 Okay, you can't set the l, you get 3f divided by ml.

01:17 Okay, so i'm not playing the numbers, the force is 14 .7, and then the mass is 0 .63.

01:27 The length is 1 .24 meters.

01:30 And you calculate you get 56 .5 radiance per second square.

01:38 And so the acm is in this case is l over 2 times alpha.

01:45 So we get 1 .24 divided by 2 times 56 .5 and you get 35 .0 meters per second square.

01:57 Okay, so this is the answer for part a.

02:00 Then in part b we need to find the horizontal force the hinge exits okay so looking at the rod okay the force is acting at the bottom 14 .7 newtons and then at the hinge there's a force acting on it so we know that the acm is 35 meters per second square so i guess the force in this case is going to be to be to the right.

02:49 Because 14 .7 divided by 0 .63 is around 29 so or less.

02:59 So with acm equals to 35, the horizontal force of hinge exits on the rod will be going to the right.

03:07 Okay, using newton's second law...

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