Physics

PHY 1301, Physics I 1

Course Learning Outcomes for Unit V Upon completion of this unit, students should be able to:

4. Apply the concept of momentum conservations to daily life. 4.1 Investigate the momentum conservation in simple pendulum motion.

5. Identify the total mechanical energy conservation.

5.1 Explain the total energy conservation in simple harmonic motion. 5.2 Relate the mass and speed of the pendulum to kinetic energy and potential energy.

Course/Unit Learning Outcomes

Learning Activity

4.1

Unit Lesson Chapter 9 Chapter 10 Unit V Problem Solving

5.1

Unit Lesson Chapter 9 Chapter 10 Unit V Problem Solving

5.2

Unit Lesson Chapter 9 Chapter 10 Unit V Problem Solving

Required Unit Resources Chapter 9: Rotational Dynamics, pp. 223–256 Chapter 10: Simple Harmonic Motion and Elasticity, pp. 257–288

Unit Lesson

Torque

Torque (T) is the magnitude of the force (F) multiplied by the lever arm (d) in a rotational motion: T = Fd. Its unit is the Joule (J), and d is the perpendicular distance between the line of action and the axis of rotation. See Figure 9.3 on page 225 in the textbook. The direction of the torque is positive with a counterclockwise rotation about the axis. On the other hand, the direction of the torque is negative with a clockwise rotation about the axis. The SI unit is Newton meter or Joule.

UNIT V STUDY GUIDE

Simple Harmonic Motion With Application

Magnitude of Torque = Magnitude of the Force x Lever Arm

PHY 1301, Physics I 2

UNIT x STUDY GUIDE

Title

The torque also can be expressed as the moment of inertia multiplied by angular acceleration for a rigid body rotating about a fixed axis. The moment of inertia varies along with the object’s shape and the location of the axis. See Table 9.1 on page 237 in the textbook. The moment of inertia depends on the geometrical structure of the object as well as the location of the rotational axis.

The SI unit of the moment of inertia is kg per meter squared. The SI unit of angular acceleration is radian per second squared.

Equilibrium Conditions for Rigid Bodies

In order to exhibit a static state of motion on an object in two-dimensional coordinates, the net force must be absent and the net torque must be zero. That is, the vertical component of the force is zero, the horizontal component of the force is zero, and the sum of the clockwise torques is the same as the sum of the counterclockwise torques. In this way, the object shows neither translational or linear motion nor rotational motion. These equilibrium conditions are essential and significant to build stable structures such as bridges, apartment buildings, and double-decker buses. In addition, they are important concepts to construct engineering equipment including simple machines such as levers, wedges, inclines, pullies, screws, wheels and axles and others (Nave, n.d.). For a more detailed explanation, please visit this Web page about "Simple Machines".

Angular Momentum Conservation

When an object rotates around an axis, the angular momentum (L) is the moment of inertia (I) multiplied by the angular velocity (w) of the object: L = Iw.

The unit of angular velocity is in radian per second, and the unit of the angular momentum is in kg m2/s. If there is no external torque in the system, the total angular momentum is conserved. It is the analogous concept of linear momentum conservation. You might have noticed that an ice skater spins more slowly when his or her arms are extended and faster when his or her arms are drawn in. See Figure 9.23 on page 244 and Example 14 on page 245 in the textbook. A satellite is orbiting around the Earth. The major force exerted on the satellite is the gravitational force of the Earth. The direction of this force points to the center of the Earth at any instant. Also, it passes through the axis about which the satellite rotates. There is no torque due to the gravitational force, and thus the angular momentum of the satellite remains constant.

Sample Question 1: Suppose the length of a wrench is 0.2 meters and the applied force is 100 N. If the force is exerted perpendicular to the wrench, what is the torque? If you apply the force with an angle, do you think that the torque will be decreased or increased? Solution: Use the formula, T = Fd. Here, F is given as 100 N, and d is 0.2 meters. So, the produced torque is 100 x 0.2 = 20 J. If you apply the force with an angle, the torque will be less than 20 J because the length of the lever arm will be shortened.

Net External Torque = Moment of Inertia x Angular Acceleration

Angular Momentum = Moment of Inertia x Angular Velocity

PHY 1301, Physics I 3

UNIT x STUDY GUIDE

Title

Angular momentum is an important physical concept because it is conserved in the universe. Most astronomical objects—planets, moons, stars, and galaxies—have this property because they rotate and revolve. Angular momentum is the tendency of a body to keep spinning or moving in a circle. It can be transferred but cannot be created or destroyed. For instance, in the sun-Earth system, the Earth circles around the sun, and the magnitude of the angular momentum, (L), can be expressed in the formula L = mvr. Here, m is the mass of Earth, v is the orbital velocity, and r is the distance between the sun and the Earth. L must be conserved; therefore, the constant value v increases as r decreases, and vice versa, with a fixed mass m. That is, v is inversely proportional to r. Similarly, the primitive giant gas cloud spins faster as more contractions occur to conserve angular momentum. The cloud flattens along the rotational axis as time passes. The center part contracts into a ball of hot gas and dust that is a protosun. The sun evolved from the protosun. Planets are formed by accretions of gases outside the sun.

The Origin of the Solar System

A favorable formation theory of our solar system is the nebular hypothesis. Immanuel Kant and Pierre Laplace postulated it for the first time in the 18th century. They hypothesized that about a billion years ago, a giant cloud of interstellar gas and dust existed in a state of balance between gravitational force and gas pressure. This giant cloud collapsed when gravity overcame pressure because of perturbation. Here, the directions between gravity and pressure are the opposite. Gravity points inward, toward collapse. Gas pressure points outward, toward expansion. If a perturbation like a supernova blast were given, gravity would have to be stronger than the inner gas pressure and could have been a trigger to contract the nebula. This giant nebula spins while it collapses to conserve angular momentum. The sun evolved from the protosun. Planets are formed by accretions of gases outside the sun. This is the basic idea of the nebula theory, but there is a shortage of evidence to explain clearly why the current sun has such a slow rotation (about a day per a rotation), unlike the primitive sun’s faster rotation. A possible solution to this mystery is that the angular momentum was transferred outside by solar wind or magnetic flux; however, exact mechanisms are not clearly understood. Even with this flaw, the nebula theory can explain the outline features of our solar system well. The advantage of the nebular theory is that it describes a basic big picture of the observed solar system although it does not perfectly explain all the details. There is observational evidence to support the theory. First, almost all planets lie in the same orbital planes as the sun. Second, the directions of the revolution/rotation of the planets are the same. Third, the rotational axes of the planets are nearly perpendicular to the orbital plane. Last, other star formation regions have been observed like the Eagle Nebula (M16), Trapezium (the central region of the Orion Nebula), the Lagoon Nebula, and Nebula NGC 604 (M33). Basic features can be observed about the planets in our solar system.

 Each planet is relatively isolated in space.

 The orbits of the planets are roughly circular with the exception of Mercury. The majority of the planets lie in the same plane, except Mercury.

 All planets revolve around the sun in one direction. It is counterclockwise as viewed from above Earth’s North Pole and the same direction as the sun rotates on its axis.

 There are differences between the terrestrial and Jovian planets. Terrestrial planets are characterized by high densities, moderate atmospheres, slow rotation, and few or no moons. By contrast, Jovian planets have low densities, thick atmospheres, rapid rotation, and many moons with rings.

PHY 1301, Physics I 4

UNIT x STUDY GUIDE

Title

Hooke’s Law

When a small amount of force applies to matter, the shape of matter changes a bit; however, it comes back to the original shape once the applied force is removed. This is elasticity. Robert Hooke (1635-1703) discovered a law of elasticity (Waggoner, n.d.). The restoring force of an ideal spring is F = -kx, where k is the spring constant and x is the displacement of the spring from its original length. The spring constant k [N/m] is often called the stiffness of the spring. As k is larger, the stiffness of the spring increases. Remember Newton’s action-reaction law. The restoring force comes from the applied force, F = kx. The applied force is proportional to the displacement.

Simple Harmonic Motion

Simple harmonic motion is a periodic or vibrational motion where Hooke’s law works. That is, the restoring force is proportional to the displacement. Examples include the motion of a pendulum, the motion of a spring, and the motion of swings in the park. In addition, the simple harmonic motion shows a sinusoidal wave in the plot between the displacement and time. A wave can be characterized by amplitude A, wavelength λ, frequency f, period T, and wave speed V. The amplitude A of a periodic wave is the maximum vertical distance from an undisturbed position. It is the distance between a crest, or the highest point, of the wave pattern and the undisturbed position. It is also the distance between a trough, or the lowest point, of the wave pattern and the undisturbed position. The size of amplitude A is related to the carrying energy. The period T is the time to complete one oscillation cycle, which is to travel one wavelength. The frequency f is the number of cycles per second, or just the reciprocal of T: f = 1 / T. The wave speed V is the wavelength λ divided by the period T: v = λ / T = f λ. The displacement x can be expressed as the product of the amplitude A and cos wt. Here, w is the angular frequency and is 2π f. The maximum velocity is the product of the amplitude A and the angular frequency w. The maximum acceleration is the product of the amplitude A and the angular frequency w squared. The angular frequency of oscillation is related to the mass of the object and the spring constant. This property can be used to design the body-mass measurement device in the space stations. For a more detailed explanation, see Example 6 on page 265 in the textbook.

Sample Question 2: An ideal spring stretched 0.2 meters with k = 50 N/m. What is the applied force? Solution: From Hooke’s law, F = kx = 50 x 0.2 = 10 N.

A depiction of Hooke’s law

Applied Force = Spring Constant x Displacement

Restoring Force = -Applied Force

PHY 1301, Physics I 5

UNIT x STUDY GUIDE

Title

The elastic potential energy due to simple harmonic motion is defined as the product of spring constant and the square of the displacement divided by two. The SI unit of elastic potential energy is the Joule. Therefore, based on what we have learned in the previous units, the total mechanical energy of a system is translational kinetic energy plus rotational kinetic energy plus gravitational potential energy plus elastic potential energy.

References

Nave, R. (n.d.). Simple machines. HyperPhysics. http://hyperphysics.phy- astr.gsu.edu/hbase/Mechanics/simmac.html#c1

Waggoner, B. (n.d.). Robert Hooke (1635–1703). University of California Museum of Paleontology.

http://www.ucmp.berkeley.edu/history/hooke.html

Learning Activities (Nongraded) Nongraded Learning Activities are provided to aid students in their course of study. You do not have to submit them. If you have questions, contact your instructor for further guidance and information.

1. Solve questions 82 and 83 pp. 255 and 256 in the textbook. 2. Solve questions 93 and 94 on p. 288 in the textbook.