A game of tug-o-war is played … but with a twist (ha!). Each team has its own rope attached to a merry-go-round. One team pulls clockwise, the other counterclockwise. Each pulls at a different point and with a different force, as shown. Who wins?
By how much? That is, what is the net torque?
Assume that the merry-go-round is weighted down with a large pile of steel plates. It is so massive that it has a moment of inertia of . What is its angular acceleration?
How long will it take the merry-go-round to spin around once completely?
You have two coins; one is a standard U.S. quarter, and the other is a coin of equal mass and size, but with a hole cut out of the center. Which coin has a higher moment of inertia?
Which coin would have the greater angular momentum if they are both spun at the same angular velocity?
A wooden plank is balanced on a pivot, as shown below. Weights are placed at various places on the plank.Consider the torque on the plank caused by weight . What force, precisely, is responsible for this torque?
What is the magnitude (value) of this force, in Newtons?
What is the moment arm of the torque produced by weight ?
What is the magnitude of this torque, in ?
Repeat parts (a – d) for weights and .
Calculate the net torque. Is the plank balanced? Explain.
A star is rotating with a period of 10.0 days. It collapses with no loss in mass to a white dwarf with a radius of of its original radius. What is its initial angular velocity?
What is its angular velocity after collapse?
For a ball rolling without slipping with a radius of , a moment of inertia of , and a linear velocity of calculate the following: The angular velocity.
The rotational kinetic energy.
The angular momentum.
The torque needed to double its linear velocity in sec.
A merry-go-round consists of a uniform solid disc of and a radius of . A single person stands on the edge when it is coasting at revolutions /sec. How fast would the device be rotating after the person has walked toward the center. (The moments of inertia of compound objects add.)
In the figure we have a horizontal beam of length, , pivoted on one end and supporting on the other. Find the tension in the supporting cable, which is at the same point at the weight and is at an angle of degrees to the vertical. Ignore the weight of the beam.
Two painters are on the fourth floor of a Victorian house on a scaffold, which weighs . The scaffold is long, supported by two ropes, each located from the end of the scaffold. The first painter of mass is standing at the center; the second of mass, , is standing from one end. Draw a free body diagram, showing all forces and all torques. (Pick one of the ropes as a pivot point.)
Calculate the tension in the two ropes.
Calculate the moment of inertia for rotation around the pivot point, which is supported by the rope with the least tension. (This will be a compound moment of inertia made of three components.)
Calculate the instantaneous angular acceleration assuming the rope of greatest tension breaks.
A horizontal beam. in length has a weight on the end. It is supported by a cable, which is connected to the horizontal beam at an angle of degrees at from the wall. Further support is provided by the wall hinge, which exerts a force of unknown direction, but which has a vertical (friction) component and a horizontal (normal) component. Find the tension in the cable.
Find the two components of the force on the hinge (magnitude and direction).
Find the coefficient of friction of wall and hinge.
On a busy intersection a beam of is connected to a post at an angle upwards of degrees to the horizontal. From the beam straight down hang a sign from the post and a signal light at the end of the beam. The beam is supported by a cable, which connects to the beam from the post at an angle of degrees measured from the beam; also by the hinge to the post, which has horizontal and vertical components of unknown direction. Find the tension in the cable.
Find the magnitude and direction of the horizontal and vertical forces on the hinge.
Find the total moment of inertia around the hinge as the axis.
Find the instantaneous angular acceleration of the beam if the cable were to break.
There is a uniform rod of mass of length . It has a mass of at one end. It is attached to the ceiling from the end with the mass. The string comes in at a degree angle to the rod. Calculate the total torque on the rod.
Determine its direction of rotation.
Explain, but don’t calculate, what happens to the angular acceleration as it rotates toward a vertical position.
The medieval catapult consists of a beam with a heavy ballast at one end and a projectile of at the other end. The pivot is located from the ballast and a force with a downward component of is applied by prisoners to keep it steady until the commander gives the word to release it. The beam is long and the force is applied from the projectile end. Consider the situation when the beam is perfectly horizontal. Draw a free-body diagram labeling all torques.
Find the mass of the ballast.
Find the force on the horizontal support.
How would the angular acceleration change as the beam moves from the horizontal to the vertical position. (Give a qualitative explanation.)
In order to maximize range at what angle should the projectile be released?
What additional information and/or calculation would have to be done to determine the range of the projectile?
Answers to Selected Problems
.
a. b.
c.
d.
e.
a. True, all rotate for which is 24 hours, b. True, and
f. True, is the same
g. and
h. True, &
i. True, &
a. b.
c.
d. Force applied perpendicular to radius allows
e. , f. and
.
Moment of inertia at the end at the center , angular momentum, and torque, change the in the same way
.
Lower
Iron ball
a. team b.
c.
d.
a. Coin with the hole b. Coin with the hole
a. weight b.
c. plank’s length left of the pivot
d. ,
e. Ba. weight, Bb. , Bc. plank’s length left of the pivot, Bd. , Ca. weight, Cb. , Cc. plank’s length right of the pivot, Cd. , f) , g) no, net torque doesn’t equal zero
a. b.
a. b.
c.
d.
b. c.
d.
a. b.
c.
a. b. toward beam, down
c.
d.
a. 20.
a. c.
d. angular acc goes down as arm moves to vertical
Chapter 11: Simple Harmonic Motion Version 2
The Big Idea
The development of devices to measure time, like the pendulum, led to the analysis of periodic motion. Such motion repeats itself in equal intervals of time (called periods) and is also referred to as harmonic motion. When an object moves back and forth over the same path in harmonic motion it is said to be oscillating. If the distance such an object travels in one oscillation remains constant, it is called simple harmonic motion (SHM). A grandfather clock’s pendulum and the quartz crystal in a modern watch are examples of SHM.
Key Concepts
The oscillating object does not lose any energy in SHM. Friction is assumed to be zero.
In harmonic motion there is always a restorative force, which attempts to restore the oscillating object to its equilibrium position. The restorative force changes during an oscillation and depends on the position of the object. In a spring the force is given by Hooke’s Law: ; in a pendulum it is the component of gravity along the path.
Objects in simple harmonic motion do not obey the “Big Three�
� equations of motion because the acceleration is not constant. As a spring compresses, the force (and hence the acceleration) increases. Similarly, as a pendulum swings, the tangential component of the force of gravity changes. The equations of motion for SHM are given in the Key Equations section.
The period, , is the amount of time needed for the harmonic motion to repeat itself, or for the object to go one full cycle. In SHM, is the time it takes the object to return to its exact starting point and starting direction.
The frequency, is the number of cycles an object goes through in 1 second. Frequency is measured in Hertz (Hz). 1 Hz = 1 cycle per sec.
The amplitude, , is the distance from the equilibrium (or center) point of motion to either its lowest or highest point (end points). The amplitude, therefore, is half of the total distance covered by the oscillating object. The amplitude can vary in harmonic motion, but is constant in SHM.
The kinetic energy and the speed are at a maximum at the equilibrium point, but the potential energy and restorative force is zero there.
At the end points the potential energy is at a maximum, while the kinetic energy and speed are zero. However at the end points the restorative force and acceleration are at a maximum.
In SHM since energy is conserved, often, the most fruitful method of calculating position and velocity is to set the total energy equal to the sum of kinetic and potential energies. Similarly force and acceleration are best calculated by using .
Key Equations and Definitions
Examples
Example 1
Question: The effective of a diving board is (we say effective because it bends in the direction of motion instead of stretching like a spring, but otherwise behaves the same). A pudgy diver is bouncing up and down at the end of the diving board. The y vs. t graph is shown below.
a) What is the distance between the lowest and the highest point of oscillation?
b) What is the y-position and velocity of the diver at ?
c) What is the diver's mass?
d) Write the sinusoidal equation of motion for the diver.
Solution:
a) As we can see from the graph the highest point is 2m and the lowest point is . Therefore the distance is
b) To find the y-position we will use the equation First we must solve for the frequency. We know that From the graph we know that the period is 2 seconds, so the frequency is hz. All we need to do now is plug in the values to find the position at .
To find the velocity we take the equation and plug in the known values.
Despite the fact that we have a negative value for the displacement it makes sense that we would get a positive velocity because, as we can see from the graph, the diving board is still moving down at .
c) To find the diver's mass we will use the equation and solve for . Then it is a simple matter to plug in the known values to get the mass. Now we plug in what we know.
d) To get the sinusoidal equation we must first decide whether it is a cosine graph or a sine graph. Then we must find the amplitude (A), vertical shift (D), horizontal shift (C), and period (B). Cosine is easier in this case so we will work with it instead of sine. As we can see from the graph, the amplitude is 2, the vertical shift is 0, and the horizontal shift is . We solved for the period already. Therefore, we can write the sinusoidal equation of this graph.
SHM Problem Set
While treading water, you notice a buoy way out towards the horizon. The buoy is bobbing up and down in simple harmonic motion. You only see the buoy at the most upward part of its cycle. You see the buoy appear 10 times over the course of one minute. What is the force that is leading to simple harmonic motion?
What is the period and frequency of its cycle? Use the proper units.
A rope can be considered as a spring with a very high spring constant so high, in fact, that you don’t notice the rope stretch at all before it “pulls back.” What is the of a rope that stretches by when a weight hangs from it?
If a boy of hangs from the rope, how far will it stretch?
If the boy kicks himself up a bit, and then is bouncing up and down ever so slightly, what is his frequency of oscillation? Would he notice this oscillation? If so, how? If not, why not?
If a mass attached to a spring oscillates 4.0 times every second, what is the spring constant of the spring?
A horizontal spring attached to the wall is attached to a block of wood on the other end. All this is sitting on a frictionless surface. The spring is compressed . Due to the compression there is of energy stored in the spring. The spring is then released. The block of wood experiences a maximum speed of . Find the value of the spring constant.
Find the mass of the block of wood.
What is the equation that describes the position of the mass?
What is the equation that describes the speed of the mass?
Draw three complete cycles of the block’s oscillatory motion on an vs. graph.
Give some everyday examples of simple harmonic motion.
Why doesn’t the period of a pendulum depend on the mass of the pendulum weight? Shouldn’t a heavier weight feel a stronger force of gravity?
The pitch of a Middle note on a piano is . This means when you hear this note, the hairs in your ears wiggle back and forth at this frequency. What is the period of oscillation for your ear hairs?
What is the period of oscillation of the struck wire within the piano?
The effective of the diving board shown here is . (We say effective because it bends in the direction of motion instead of stretching like a spring, but otherwise behaves the same.) A pudgy diver is bouncing up and down at the end of the diving board, as shown. The vs graph is shown below. What is the distance between the lowest and highest points of oscillation?
What is the position of the diver at times and ?
Estimate the man’s period of oscillation.
What is the diver’s mass?
Write the sinusoidal equation of motion for the diver.
The Sun tends to have dark, Earth-sized spots on its surface due to kinks in its magnetic field. The number of visible spots varies over the course of years. Use the graph of the sunspot cycle above to answer the following questions. (Note that this is real data from our sun, so it doesn’t look like a perfect sine wave. What you need to do is estimate the best sine wave that fits this data.) Estimate the period in years.
When do we expect the next “solar maximum?”
The pendulum of a small clock is long. How many times does it go back and forth before the second hand goes forward one second?
On the moon, how long must a pendulum be if the period of one cycle is one second? The acceleration of gravity on the moon is one sixth that of Earth.
A spider of walks to the middle of her web. The web sinks by due to her weight. You may assume the mass of the web is negligible. If a small burst of wind sets her in motion, with what frequency will she oscillate?
How many times will she go up and down in one s? In ?
How long is each cycle?
Draw the vs graph of three cycles, assuming the spider is at its highest point in the cycle at .
A mass on a spring on a frictionless horizontal surface undergoes SHM. The spring constant is and the mass is . The initial amplitude is . At the point of release find: the potential energy
the horizontal force on the mass
the acceleration as it is released
As the mass reaches the equilibrium point find: the speed of the mass
the horizontal force on the mass
the acceleration of the mass
At a point from the equilibrium point find: the potential and kinetic energy
the speed of the mass
the force on the mass
the acceleration of the mass
Find the period and frequency of the harmonic motion.
A pendulum with a string of and a mass of is given an initial amplitude by pulling it upward until it is at a height of more than when it hung v
ertically. This is point . When it is allowed to swing it passes through point at a height of above the equilibrium position, the latter of which is called point . Draw a diagram of this pendulum motion and at points and draw velocity and acceleration vectors. If they are zero, state that also.
At point calculate the potential energy.
At point calculate the speed of the mass.
At point calculate the speed of the mass.
If the string were to break at points and draw the path the mass would take until it hits ground for each point.
Find the tension in the string at point .
Find the tension in the string at point .
Find the period of harmonic motion.
Answers to Selected Problems
a. Buoyant force and gravity b.
a. b.
c. , no,
CK-12 People's Physics Book Version 2 Page 10
