You throw a lump of clay with a speed of at a bowling ball hanging from a vertical rope. The bowling ball swings up to a height of compared to its initial height. Was this an elastic collision? Justify your answer.
The bullet shown below is traveling to the right with a speed of . A block is hanging from the ceiling from a rope in length. What is the maximum height that the bullet-block system will reach, if the bullet embeds itself in the block?
What is the maximum angle the rope makes with the vertical after the collision?
You are playing pool and you hit the cue ball with a speed of at the ball (which is stationary). Assume an elastic collision and that both balls are the same mass. Find the speed and direction of both balls after the collision, assuming neither flies off at any angle.
A golf ball with a speed of collides elastically head-on with a pool ball at rest. Find the speed and direction of both balls after the collision.
Ball is traveling along a flat table with a speed of , as shown below. Ball , which has the same mass, is initially at rest, but is knocked off the 1.5m high table in an elastic collision with Ball . Find the horizontal distance that Ball travels before hitting the floor.
Manrico and Leonora are figure skaters. They are moving toward each other. Manrico’s speed is ; Leonora’s speed is . When they meet, Leonora flies into Manrico’s arms. With what speed does the entwined couple move?
In which direction are they moving?
How much kinetic energy is lost in the collision?
Aida slides down a high hill on a frictionless sled (combined mass . At the bottom of the hill, she collides with Radames on his sled (combined mass . The two children cling together and move along a horizontal plane that has a coefficient of kinetic friction of . What was Aida’s speed before the collision?
What was the combined speed immediately after collision?
How far along the level plane do they move before stopping?
A pile driver lifts a mass a vertical distance of in sec. It uses of supplied power to do this. How much power was used in actually lifting the mass?
What is the efficiency of the machine? (This is the ratio of power used to power supplied.)
The mass is dropped on a pile and falls . If it loses on the way down to the ground due to air resistance, what is its speed when it hits the pile?
Investigating a traffic collision, you determine that a fast-moving car (mass ) hit and stuck to a second car (mass ), which was initially at rest. The two cars slid a distance of over rough pavement with a coefficient of friction of before coming to a halt. What was the speed of the first car? Was the driver above the posted speed limit?
Force is applied in the direction of motion to a cart on a frictionless surface. The motion is along a straight line and when , then and . (The displacement and velocity of the cart are initially zero.) Look at the following graph: What is the change in momentum during the first sec?
What is the change in velocity during the first sec?
What is the acceleration at sec?
What is the total work done on the cart by the force from sec?
What is the displacement after sec?
Force is applied in the direction of motion to a cart on a frictionless surface. The motion is along a straight line and when and . look at the following graph: What is the acceleration of the cart when the displacement is ?
What work was done on the cart between and ?
What is the total work done on the cart between ?
What is the speed of the cart at ?
What is the impulse given the cart by the force from ?
What is the speed at ?
How much time elapsed from when the cart was at to when it got to ?
You are to design an experiment to measure the average force an archer exerts on the bow as she pulls it back prior to releasing the arrow. The mass of the arrow is known. The only lab equipment you can use is a meter stick. Give the procedure of the experiment and include a diagram with the quantities to be measured shown.
Give sample calculations using realistic numbers.
What is the single most important inherent error in the experiment?
Explain if this error would tend to make the force that it measured greater or lesser than the actual force and why.
Molly eats a power bar before the big pole vault. The bar’s energy content comes from changing chemical bonds from a high to a low state and expelling gases. However, % of the bar’s energy is lost expelling gases and % is needed by the body for various biological functions. How much energy is available to Molly for the run?
Energy losses due to air resistance and friction on the run are , Molly’s increased heart rate and blood pressure use of the available energy during the run. What top speed can the Molly expect to attain?
The kinetic energy is transferred to the pole, which is “compressed” like a spring of ; air resistance energy loss on the way up is , and as she crosses the bar she has a horizontal speed of . If Molly rises to a height equal to the expansion of the pole what is that height she reaches?
On the way down she encounters another of air resistance. How much heat in the end is given up when she hits the dirt and comes to a stop?
A new fun foam target on wheels for archery students has been invented. The arrow of mass, , and speed, , goes through the target and emerges at the other end with reduced speed, . The mass of the target is . Ignore friction and air resistance. What is the final speed of the target?
What is the kinetic energy of the arrow after it leaves the target?
What is the final kinetic energy of the target?
What percent of the initial energy of the arrow was lost in the shooting?
Answers to Selected Problems
.
.
.
.
.
a. b.
a. @ @ @ @ and F; @ b.
c. Yes, it makes the loop
a. c. No, the baby will not clear the hill.
a. b. Spring has compressed length of
.
a. b.
a. b.
each
a. b.
a. b. above the spring
%
.
a. b. 5.
same direction as the cue ball and
a. b. Leonora’s
c.
a. b.
c.
a. b.
c.
.
.
.
a. b.
c.
d.
a. b.
c.
d. %
Chapter 10: Rotational Motion Version 2
The Big Idea
In the chapter on centripetal forces, we learned that in some situations, objects move in circles. The purpose of this chapter is to describe and formalize such motion. The fundamental physics behind it is based on the conservation of angular momentum. This vector quantity is the product of rotational velocity and rotational inertia. In any closed system (including the universe) the quantity of angular momentum is fixed. Angular momentum can be transferred from one body to another, but cannot be lost or gained. If a system has its angular momentum changed from the outside it is caused by a torque. Torque is a force applied at a distance from the center of rotation.
Rotational motion has many analogies to linear motion. By studying it in this framework, we can make use of many of our previous results. In fact, most of rotational motion can be understood by looking at the following figure and applying results from previous chapters.
Formalizing Rotational Motion
Figure 10.1
Illustration of Rotational Motion
Key Concepts
To determine the rotation axis, wrap your right hand's fingers in the direction of rotation and your thumb will point along the axis (see figure).
When something rotates in a circle, it moves through a position angle that runs from to radians and starts over again at . The p
hysical distance it moves is called the path length. If the radius of the circle is larger, the path length traveled is longer. According to the arc length formula , the path length traveled by something at radius through an angle is:
Just like the linear velocity is the rate of change of distance, angular velocity, usually called , is the rate of change of . The direction of angular velocity is either clockwise or counterclockwise. Analogously, the rate of change of is the angular acceleration .
The linear velocity and linear acceleration of rotating object also depend on the radius of rotation, which is called the moment arm (See figure) If something is rotating at a constant angular velocity, it moves more quickly if it is farther from the center of rotation. For instance, people at the Earth’s equator are moving faster than people at northern latitudes, even though their day is still 24 hours long – this is because they have a greater circumference to travel in the same amount of time. According to [1],
Alternatively, we could derive [2] by setting the time to travel a path length equal to the circumference, at speed equal to the time it takes to travel one full angular revolution, at angular velocity .
In exactly the same fashion we can derive the fact that angular acceleration is related to linear acceleration in the following way:
Note that angular acceleration is not the same as centripetal acceleration, which always points toward the center. Angular acceleration is always in the direction or against the direction of angular velocity. The linear acceleration associated with it points along instantaneous velocity.
Since the algebra is identical, under constant angular acceleration we can have the big three equations for circular motion.
Just as linear accelerations are caused by forces, angular accelerations are caused by torques.
Torques produce angular accelerations, but just as masses resist acceleration (due to inertia), there is an inertia that opposes angular acceleration. The measure of this inertial resistance depends on the mass, but more importantly on the distribution of the mass in a given object. The moment of inertia, , is the rotational version of mass. Values for the moment of inertia of common objects are given in problem 2. Torques have only two directions: those that produce clockwise (CW) and those that produce counterclockwise (CCW) rotations. The angular acceleration or change in would be in the direction of the torque.
Imagine spinning a fairly heavy disk. To make it spin, you don’t push towards the disk center– that will just move it in a straight line. To spin it, you need to push along the side, much like when you spin a basketball. Thus, the torque you exert on a disk to make it accelerate depends only on the component of the force perpendicular to the radius of rotation:
Many separate torques can be applied to an object. The angular acceleration produced is
The angular momentum of a spinning object is . Torques produce a change in angular momentum with time:
Spinning objects have a kinetic energy, given by .
Analogies Between Linear and Rotational Motion
In addition to [1], [2], and [3], there are other important relationships for rotational motion. These are summarized in table 1.1.
Important Rotational Motion Formulas. Equation Explanation
the centripetal acceleration of an object.
Relationship between period and frequency.
The ‘Big Three’ equations work for rotational motion too!
Rotational equivalent of
Rotational equivalent of
Angular accelerations are produced by net torques,with inertia opposing acceleration; this is the rotational analog of
The net torque is the vector sum of all the torques acting on the object. When adding torques it is necessary to subtract CW from CCW torques.
Individual torques are determined by multiplying the force applied by the perpendicular component of the moment arm
Angular momentum is the product of moment of inertia and angular velocity.
Torques produce changes in angular momentum; this is the rotational analog of
Rotational motion contributes to kinetic energy as well!
Example 1
Question: A game of tug-of-war is played...but with a twist (ha!). Each team has its own rope attached to a merry-go-round. One team pulls counterclockwise with a force of 200N. The other team pulls clockwise with a force of 400N. But there is another twist. The counterclockwise team's rope is attached 2.6m from the center of the merry go round and the clockwise team's rope is attached 1.2m from the center of the circle.
a) Who wins?
b) By how much? That is, what is the net torque?
c) 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 the angular acceleration?
d) How long will it take the merry-go-round to complete one revolution?
Solution:
a) To find out who wins, we need to find which team is pulling with the greater torque. Therefore, we will use the equation counterclockwise team: clockwise team: So the counterclockwise team wins.
b) To figure out the net torque we simply subtract the two torques. So we have a counterclockwise net torque.
c) To find the angular acceleration we use the equation Since we know both the net torque and the moment of inertia, all we have to do is plug these values in.
d) Finally, we want to know the time of one rotation. To do this we will use the equation We are only concerned with because and both equal 0. All we need to do is solve for time. Now we plug in the known values to get time.
Rotational Motion Problem Set
The wood plug, shown below, has a lower moment of inertia than the steel plug because it has a lower mass. Which of these plugs would be easier to spin on its axis? Explain.
Even though they have the same mass, the plug on the right has a higher moment of inertia (I), than the plug on the left, since the mass is distributed at greater radius.
Which of the plugs would have a greater angular momentum if they were spinning with the same angular velocity? Explain.
Here is a table of some moments of inertia of commonly found objects: Calculate the moment of inertia of the Earth about its spin axis.
Calculate the moment of inertia of the Earth as it revolves around the Sun.
Calculate the moment of inertia of a hula hoop with mass and radius .
Calculate the moment of inertia of a rod in length and mass rotating about one end.
Repeat d., but calculate the moment of inertia about the center of the rod.
Imagine standing on the North Pole of the Earth as it spins. You would barely notice it, but you would turn all the way around over 24 hours, without covering any real distance. Compare this to people standing on the equator: they go all the way around the entire circumference of the Earth every 24 hours! Decide whether the following statements are TRUE or FALSE. Then, explain your thinking. The person at the North Pole and the person at the equator rotate by radians in seconds.
The angular velocity of the person at the equator is radians per second.
Our angular velocity in San Francisco is radians per second.
Every point on the Earth travels the same distance every day.
Every point on the Earth rotates through the same angle every day.
The angular momentum of the Earth is the same each day.
The angular momentum of the Earth is .
The rotational kinetic energy of the Earth is .
The orbital kinetic energy of the Earth is , where refers to the distance from the Earth to the Sun.
You spin up some pizza dough from rest with an angular acceleration of . How many radians has the pizza dough spun through in the first seconds?
How many times has the pizza dough spun around in this time?
What is its angular velocity after seconds?
What is providing the torque that allows the angular acceleration to occur?
Calculate the moment of inert
ia of a flat disk of pizza dough with mass and radius .
Calculate the rotational kinetic energy of your pizza dough at and .
Your bike brakes went out! You put your feet on the wheel to slow it down. The rotational kinetic energy of the wheel begins to decrease. Where is this energy going?
Consider hitting someone with a Wiffle ball bat. Will it hurt them more if you grab the end or the middle of the bat when you swing it? Explain your thinking, but do so using the vocabulary of moment of inertia (treat the bat as a rod), angular momentum (imagine the bat swings down in a semi-circle), and torque (in this case, torques caused by the contact forces the other person’s head and the bat are exerting on each other).
Why does the Earth keep going around the Sun? Shouldn’t we be spiraling farther and farther downward towards the Sun, eventually falling into it? Why do low-Earth satellites eventually spiral down and burn up in the atmosphere, while the Moon never will?
If most of the mass of the Earth were concentrated at the core (say, in a ball of dense iron), would the moment of inertia of the Earth be higher or lower than it is now? (Assume the total mass stays the same.)
Two spheres of the same mass are spinning in your garage. The first is in diameter and made of iron. The second is in diameter but is a thin plastic sphere filled with air. Which is harder to slow down? Why? (And why are two spheres spinning in your garage?)
CK-12 People's Physics Book Version 2 Page 9
