Sunday, September 30, 2007
Impulse and Momentum
If momentum changes, it’s because mass or velocity change.
Most often mass doesn’t change so velocity changes and that is acceleration.
And mass x acceleration = force
Applying a force over a time interval to an object changes the momentum
Force x time interval = Impulse
Impulse = F x t or Ft = mv
Most often mass doesn’t change so velocity changes and that is acceleration.
And mass x acceleration = force
Applying a force over a time interval to an object changes the momentum
Force x time interval = Impulse
Impulse = F x t or Ft = mv
What’s Momentum ?
Momentum = mass x velocity
momentum = mv
if direction is not an important factor : . . momentum = mass x speed
So, A really slow moving truck and an extremely fast roller skate can have the same momentum.
Unit: kgm/s
momentum = mv
if direction is not an important factor : . . momentum = mass x speed
So, A really slow moving truck and an extremely fast roller skate can have the same momentum.
Unit: kgm/s
Thursday, September 20, 2007
Linear Momentum (Chapter 7)
Objectives
1. Define linear momentum and write the mathematical formula for linear momentum from memory.
2. Distinguish between the unit of force and momentum.
3. Write Newton's Second Law of Motion in terms of momentum.
4. Define impulse and write the equation that connects impulse and momentum.
5. State the Law of Conservation of Momentum and write, in vector form, the law for a system involving two or more point masses.
6. Distinguish between a perfectly elastic collision and a completely inelastic collision.
7. Apply the laws of conservation of momentum and energy to problems involving collisions between two point masses.
8. Define center of mass and center of gravity and distinguish between the two concepts.
1. Define linear momentum and write the mathematical formula for linear momentum from memory.
2. Distinguish between the unit of force and momentum.
3. Write Newton's Second Law of Motion in terms of momentum.
4. Define impulse and write the equation that connects impulse and momentum.
5. State the Law of Conservation of Momentum and write, in vector form, the law for a system involving two or more point masses.
6. Distinguish between a perfectly elastic collision and a completely inelastic collision.
7. Apply the laws of conservation of momentum and energy to problems involving collisions between two point masses.
8. Define center of mass and center of gravity and distinguish between the two concepts.
Practice Questions VI
1. Two men, Joel and Jerry, push against a wall. Jerry stops after 10 min, while Joel is able to push for 5 min longer. Compare the work against the wall they each do.
(A) Joel does 50% more work than Jerry.
(B) Jerry does 50% more work than Joel.
(C) Joel does 75% more work than Jerry.
(D) Neither of them do any work.
13. Two cars, starting from rest at the same place, travel by different routes to the same destination. One of the cars passes the other as they drive through it. Which of the following statements will be true?
(A) Joel does 50% more work than Jerry.
(B) Jerry does 50% more work than Joel.
(C) Joel does 75% more work than Jerry.
(D) Neither of them do any work.
2. A simple pendulum, consisting of a mass m and a string, swings upward, making an angle θ with the vertical. The work done by the tension force is
(A) zero.
(B) mg.
(C) mg cos theta
(C) mg sin theta
3. A simple pendulum, consisting of a mass m, is attached to the end of a 1.5 m length of string. If the mass is held out horizontally, and then released from rest, its speed at the bottom is
(A) 4.4 m/s
(B) 5.4 m/s
(C) 9.8 m/s
(D) 17 m/s
4. A 4-kg mass moving with speed 2 m/s, and a 2-kg mass moving with a speed of 4 m/s, are gliding over a horizontal frictionless surface. Both objects encounter the same horizontal force, which directly opposes their motion, and are brought to rest by it. Which statement best describes their respective stopping distances?
(A) The 4-kg mass travels twice as far as the 2-kg mass before stopping.
(B) The 2-kg mass travels twice as far as the 4-kg mass before stopping.
(C) Both masses travel the same distance before stopping.
(D) The 2 kg mass travels greater than twice as far.
5. A 4-kg mass moving with speed 2 m/s and, an otherwise identical, 2-kg mass moving with a speed of 4 m/s, are gliding over a horizontal surface with friction and are brought to rest by it. Which statement best describes their respective stopping distances?
(A) The 4-kg mass travels twice as far as the 2-kg mass before stopping.
(B) The 2-kg mass travels twice as far as the 4-kg mass before stopping.
(C) Both masses travel the same distance before stopping.
(D) The 2-kg mass travels greater than twice as far.
6. A force that Object A exerts on Object B is observed over a 10-second interval, as shown on the graph. How much work did Object A do during that 10 s?
(A) Zero
(B) 12.5 J
(C) 25 J
(D) 50 J
7. A force that Object A exerts on Object B is observed over a 10-second interval, as shown on the graph. What is the average power output of A into B?
(A) O W
(B) 1.3 W
(C) 2.5 W
(D) 5 W
8. The resultant force you exert on a shopping cart, for a 10 s period, is plotted on the graph shown. How much work did you do during this 10 s interval?
(A) Zero
(B) 12.5 J
(C) 25 J
(D) -25 J
9. The resultant force you exert on a shopping cart, for a 10 s period, is plotted on the graph, shown. Which of the following statements are true?
(A) The average power input into B is greater than zero.
(B) The average power input into A is the same in the first half as the power input in the second half.
(C) The average power equals the instantaneous power.
(D) The average power is zero.
10. How much work was required to bring the 1000-kg roller coaster from Point P to rest at Point Q at the top of the 50 m peak?
(A) 32,000 J
(B) 50,000 J
(C) 245,000 J
(D) 490,000 J
11. If the roller coaster leaves Point Q from rest, how fast is it traveling at Point R?
(A) 22.1 m/s
(B) 31.3 m/s
(C) 490 m/s
(D) 980 m/s
12. What was the total work done on you by all the forces in the universe between the time just before you awoke this morning and right now?
(A) Don't have a clue
(B) Greater than zero.
(C) Zero.
(D) Can not be calculated.
13. Two cars, starting from rest at the same place, travel by different routes to the same destination. One of the cars passes the other as they drive through it. Which of the following statements will be true?
(A) The work done by friction during the trip was the same for both
(B) The total work done on both is the same.
(C) The work done by gravity is the same on both.
(D) The work done by gravity on both is positive.
14. The work done by friction, f,
(A) equals -fd, where d is the total distance moved.
(B) equals fd, where d is the total distance.
(C) can't easily be calculated because it depends on the angle between f and d.
(D) can't easily be calculated.
15. A 102 kg man climbs a 5.0 meter high stair case at constant speed. How much work does he do?
(A) 510 J
(B) 49 J
(C) 5000 J
(D) 2500 J
15. A ball is released, from rest, at the left side of the loop-the-loop, at the height shown (h = 2R). If the radius of the loop is R and there is no friction, what vertical height does the ball rise to on the other side?
(A) Less than R
(B) R
(C) 2R
(D) Greater than R
Mechanical Energy and its Conservation
The change in total mechanical energy is the work done by non-conservative forces.
In case there aren’t any (no friction, etc.):
W(nc) = change in KE + change in PE
The total mechanical energy doesn’t change. It is
conserved in the absence of non-conservative forces.
W(nc) = change in KE + change in PE
In case there aren’t any (no friction, etc.):
W(nc) = change in KE + change in PE
The total mechanical energy doesn’t change. It is
conserved in the absence of non-conservative forces.
W(nc) = change in KE + change in PE
non-conservative forces
For non-conservative force work done does depend on
the chosen path.
We learned Wfr= -Ffr ∙ d → longer path requires more work.
Coefficient of friction may be different.
Conservative forces
A force F is acting on an object.
Gravitational force: mechanical work against it depends
just on difference in elevation not how an object is
lifted.
Elastic force: work against spring only depends on
length change.
If the work done by F to get from A to B is independent of the path, then F is a said to be a conservative force.
The force of gravity is a conservative force.
Gravitational force: mechanical work against it depends
just on difference in elevation not how an object is
lifted.
Elastic force: work against spring only depends on
length change.
Hooke's Law
F(p) = kxF(s) = -kx
k = Spring constant [N/m]
k = Spring constant [N/m]
F is not constant,
the average force = (0 + 1/2 kx)
W = 1/2 kx^2
elastic PE= 1/2 kx^2
Conservation of Energy
In a closed system energy is conserved.
The position of the blue ball ,there is the Potential Energy (PE) while the Kinetic Energy (KE) = 0.
As the blue ball is approching the purple ball position the PE is decreasing while the KE is increasing. At exactly halfway between the blue and purple ball position the PE = KE.
The position of the purple ball is where the Kinetic Energy is at its maximum while the Potential Energy (PE) = 0.
At this point, theoretically, all the PE has transformed into KE> Therefore now the KE = 19.6J while the PE = 0.
The position of the pink ball is where the Potential Energy (PE) is once again at its maximum and the Kinetic Energy (KE) = 0.
PE + KE = 0
PE = -KE
The sum of PE and KE is the total mechanical energy:
Total Mechanical Energy = PE + KE
Potential Energy
The gravitational potential energy depends on the height
of an object over some reference level.
The potential energy depends only on the position of the object.
PE = mgh
where
PE = Energy (in Joules)
m = mass (in kilograms)
g = gravitational acceleration of the earth (9.8 m/sec2)
h = height above earth's surface (in meters)
PE = Energy (in Joules)
m = mass (in kilograms)
g = gravitational acceleration of the earth (9.8 m/sec2)
h = height above earth's surface (in meters)
W(net) = KE2 - KE1Work and Kinetic Energy
•An object’s kinetic energy can be likened to the work that could be done if object were brought to rest (so, the K.E. is like potential work content)
–The moving hammer has kinetic energy and can do work on the nail .
Kinetic Energy
Kinetic Energy KE is the energy associated with the motion of an object .
Kinetic Energy exists whenever an object which has mass is in motion with some velocity. Everything you see moving about has kinetic energy.
KE = 1/2 mv(square)
where
KE = Energy (in Joules)
m = mass (in kilograms)
v = velocity (in meters/sec)
Wednesday, September 19, 2007
A Teacher applies a Force to a Wall.
W = F d
d = 0
The wall is not displaced.
Work Done is Zero when the Displacement is Zero.
A teacher applies a force to a wall and becomes exhausted. But No Work is Done.
The force exerts the brakes to stop the car.
The direction of the force is in the opposite direction that the object moves.
W = F cos 180˚ d
cos 180 ˚ = - 1
W= - F d
The Work Done on the car is negative because of the frictional force.
Frictional Force is always opposing the relative motion of the body.
A Waiter Carrying A Tray
W = F cos 90 ˚d
W = 0
No Work is Done.
•When a particular force is perpendicular to the motion, no work is done by that force.
Unit of Work
SI system: 1 Joule (J) = 1 N.m
cgs system: 1 erg = 1 dyne.cm = 10-7 J
1 calorie (cal) = 4.19 J
British units 1 foot-pound (ft.lb) = 1.36 J
cgs system: 1 erg = 1 dyne.cm = 10-7 J
1 calorie (cal) = 4.19 J
British units 1 foot-pound (ft.lb) = 1.36 J
Work done by a constant force
The work W done by a constant force on an object is
the product of the magnitude of the displacement
and the component of the force parallel to the displacement.
W = F// d
W = Fd cos (theta)
vector F = the constant force
d = the magnitude of the displacement of the object
In S.I unit, 1 Nm = 1 J
Work Done is a scalar quantity.
the product of the magnitude of the displacement
and the component of the force parallel to the displacement.
W = F// d
W = Fd cos (theta)
vector F = the constant force
d = the magnitude of the displacement of the object
In S.I unit, 1 Nm = 1 J
Work Done is a scalar quantity.
Work and Energy (Chapter 6)
Objectives
1. Distinguish between work in the scientific sense as compared to the colloquial sense.
2. Write the definition of work in terms of force and displacement and calculate the work done by a constant force when the force and displacement vectors are at an angle.
3. Use graphical analysis to calculate the work done by a force that varies in magnitude.
4. Define each type of mechanical energy and give examples of types of energy that are not mechanical.
5. State the work energy theorem and apply the theorem to solve problems.
6. Distinguish between a conservative and a nonconservative force and give examples of each type of force.
7. State the law of conservation of energy and apply the law to problems involving mechanical energy.
8. Define power in the scientific sense and solve problems involving work and power.
1. Distinguish between work in the scientific sense as compared to the colloquial sense.
2. Write the definition of work in terms of force and displacement and calculate the work done by a constant force when the force and displacement vectors are at an angle.
3. Use graphical analysis to calculate the work done by a force that varies in magnitude.
4. Define each type of mechanical energy and give examples of types of energy that are not mechanical.
5. State the work energy theorem and apply the theorem to solve problems.
6. Distinguish between a conservative and a nonconservative force and give examples of each type of force.
7. State the law of conservation of energy and apply the law to problems involving mechanical energy.
8. Define power in the scientific sense and solve problems involving work and power.
Monday, August 20, 2007
Practice Questions (Chapter 5)
1. A stone is whirled in a vertical circle on a cord. Halfway up
(A) the tension is towards the center of the circle and the net force is down.
(B) the weight is down and the net force is towards the center of the circle.
(C) the tension force is towards the center of the circle and the weight is down.
(D) the weight and tension are in the same direction.
2. A roller coaster car is on a track that forms a circular loop in the vertical plane. If the car is to just maintain contact with the track at the top of the loop, what is the minimum value for its centripetal acceleration at this point?
(A) g downward
(B) g upward
(C) 0.5 g downward
(D) 2 g upward
3. Two objects are travelling in circular orbits. Object A is travelling at twice the velocity of object B in a circle with a diameter of twice that of B. The centripetal acceleration
(A) of A and B are the same.
(B) of A is twice that B.
(C) of A is four times that of B.
(D) of A is half that of B.
4. A cannon ball is fired horizontally off a high cliff over a great distance. If air resistance can be ignored the path it follows is
(A) part of a parabola.
(B) part of a circle.
(C) a hyperbola.
(D) part of an ellipse.
5. There are four forces in nature. Which one allows you to close a door by pushing on it?
(A) Gravity
(B) Electric
(C) Nuclear
(D) Weak
6. A ball of mass m is moving in a circle with uniform speed on a horizontal surface with friction at the end of a radial metal rod. The net force is
(A) opposite the friction force.
(B) in the direction of the friction force.
(C) perpendicular to the surface.
(D) is along the rod.
7. Two bodies of equal mass are separated by a distance R. If you double the distance between them the new gravitational force will be
(A) twice the old force.
(B) half the old force.
(C) four times the old force.
(D) one fourth the old force.
(E) The same as the old force.
8. Two bodies of equal mass are separated by a distance R. If you double each mass then the new force will be
(A) twice the old force.
(B) half the old force.
(C) four times the old force.
(D) one fourth the old force.
(E) the same as the old force.
9. Two bodies of equal mass are separated by a distance R. If you double one mass then the new force will be
(A) twice the old force.
(B) half the old force.
(C) four times the old force.
(D) one fourth the old force.
(E) the same as the old force.
10. Two bodies of equal mass are separated by a distance R. If you double each mass and double the distance between them, the new force will be
(A) twice the old force.
(B) half the old force.
(C) four times the old force.
(D) one fourth the old force.
(E) the same as the old force.
11. The Earth moves faster in the winter when its closest to the Sun. This is an example of
(A) Newton's Law of Gravity.
(B) Kepler's First Law.
(C) Kepler's Second Law.
(D) Kepler's Third Law.
12. Which of the following does not belong with the others?
(A) A car speeds up from 30 mph to 40 mph.
(B) A car breaks from 20 mph to 0.
(C) A race car rounds a curve at 120 mph.
(D) A car coasts down a road at constant speed.
13. Kepler's Third Law can be used to relate the motions of the
(A) Moon and the Earth.
(B) Moon and the Space Shuttle.
(C) Earth and the Sun.
(D) Moon and the Sun.
14. If the mass of the Earth was quadrupled and nothing else was changed
(A) g would increase by a factor of 4.
(B) g would increase by a factor of 2.
(C) g would decrease by a factor of 1/2.
(D) g would decrease by a factor of 1/4.
15. A planet with twice the mass and twice its radius has a value of g
(A) four times that of Earth.
(B) two times that of Earth.
(C) 1/2 that of Earth.
(D) 1/4 that of Earth.
16. Four equal masses are arranged in a perfect square. The direction of the force on the one in the lower left hand corner
(A) depends on the value of the mass.
(B) is at 45 degrees.
(C) is at 225 degrees.
(D) zero.
17. The speed of a satellite in orbit does not depend on
(A) its mass.
(B) the mass of the Earth.
(C) its distance above the Earth's surface.
(A) the tension is towards the center of the circle and the net force is down.
(B) the weight is down and the net force is towards the center of the circle.
(C) the tension force is towards the center of the circle and the weight is down.
(D) the weight and tension are in the same direction.
2. A roller coaster car is on a track that forms a circular loop in the vertical plane. If the car is to just maintain contact with the track at the top of the loop, what is the minimum value for its centripetal acceleration at this point?
(A) g downward
(B) g upward
(C) 0.5 g downward
(D) 2 g upward
3. Two objects are travelling in circular orbits. Object A is travelling at twice the velocity of object B in a circle with a diameter of twice that of B. The centripetal acceleration
(A) of A and B are the same.
(B) of A is twice that B.
(C) of A is four times that of B.
(D) of A is half that of B.
4. A cannon ball is fired horizontally off a high cliff over a great distance. If air resistance can be ignored the path it follows is
(A) part of a parabola.
(B) part of a circle.
(C) a hyperbola.
(D) part of an ellipse.
5. There are four forces in nature. Which one allows you to close a door by pushing on it?
(A) Gravity
(B) Electric
(C) Nuclear
(D) Weak
6. A ball of mass m is moving in a circle with uniform speed on a horizontal surface with friction at the end of a radial metal rod. The net force is
(A) opposite the friction force.
(B) in the direction of the friction force.
(C) perpendicular to the surface.
(D) is along the rod.
7. Two bodies of equal mass are separated by a distance R. If you double the distance between them the new gravitational force will be
(A) twice the old force.
(B) half the old force.
(C) four times the old force.
(D) one fourth the old force.
(E) The same as the old force.
8. Two bodies of equal mass are separated by a distance R. If you double each mass then the new force will be
(A) twice the old force.
(B) half the old force.
(C) four times the old force.
(D) one fourth the old force.
(E) the same as the old force.
9. Two bodies of equal mass are separated by a distance R. If you double one mass then the new force will be
(A) twice the old force.
(B) half the old force.
(C) four times the old force.
(D) one fourth the old force.
(E) the same as the old force.
10. Two bodies of equal mass are separated by a distance R. If you double each mass and double the distance between them, the new force will be
(A) twice the old force.
(B) half the old force.
(C) four times the old force.
(D) one fourth the old force.
(E) the same as the old force.
11. The Earth moves faster in the winter when its closest to the Sun. This is an example of
(A) Newton's Law of Gravity.
(B) Kepler's First Law.
(C) Kepler's Second Law.
(D) Kepler's Third Law.
12. Which of the following does not belong with the others?
(A) A car speeds up from 30 mph to 40 mph.
(B) A car breaks from 20 mph to 0.
(C) A race car rounds a curve at 120 mph.
(D) A car coasts down a road at constant speed.
13. Kepler's Third Law can be used to relate the motions of the
(A) Moon and the Earth.
(B) Moon and the Space Shuttle.
(C) Earth and the Sun.
(D) Moon and the Sun.
14. If the mass of the Earth was quadrupled and nothing else was changed
(A) g would increase by a factor of 4.
(B) g would increase by a factor of 2.
(C) g would decrease by a factor of 1/2.
(D) g would decrease by a factor of 1/4.
15. A planet with twice the mass and twice its radius has a value of g
(A) four times that of Earth.
(B) two times that of Earth.
(C) 1/2 that of Earth.
(D) 1/4 that of Earth.
16. Four equal masses are arranged in a perfect square. The direction of the force on the one in the lower left hand corner
(A) depends on the value of the mass.
(B) is at 45 degrees.
(C) is at 225 degrees.
(D) zero.
17. The speed of a satellite in orbit does not depend on
(A) its mass.
(B) the mass of the Earth.
(C) its distance above the Earth's surface.
Tuesday, July 24, 2007
Satellites and "Weightlessness"
A satellite is put into orbit by accelerating it to a sufficient high tangential speed with the use of rockets.
If the speed is too high, the spacecraft will not be confined by the Earth's gravity and will escape, never to return.
If the speed is too low, it will return to Earth.
Satellite are usually put into circular (or nearly circular) orbits, because such orbits require the least takeoff speed.
this image is from www.howstuffworks.com
If a satellite stopped moving, it would fall directly to Earth.
But at the very high speed a satellite has, it would quickly fly out into space, if it weren't for the gravitational force of the Earth pulling it into orbit.
In fact, a satellite is falling (accelerating toward Earth), but its high tangential speed keeps it from hitting Earth.
If the speed is too high, the spacecraft will not be confined by the Earth's gravity and will escape, never to return.
If the speed is too low, it will return to Earth.
Satellite are usually put into circular (or nearly circular) orbits, because such orbits require the least takeoff speed.
this image is from www.howstuffworks.com
But at the very high speed a satellite has, it would quickly fly out into space, if it weren't for the gravitational force of the Earth pulling it into orbit.
In fact, a satellite is falling (accelerating toward Earth), but its high tangential speed keeps it from hitting Earth.
Monday, July 23, 2007
Gravity Near the Earth's Surface: Geophysical Application
When we apply the gravitaional force to the gravitational force between the Earth and an object at its surface,
m(1) becomes the mass of the Earth m(E) .
m(2) becomes tge mass of the object m.
r becomes the distance of the object from the Earth's centre = radius of the earth
The force of gravity due to the Earth is the weight of the object mg.
Wednesday, July 18, 2007
Thursday, July 12, 2007
Nonuniform Circular Motion
Any object moving in a circle has a centripetal acceleration,
directed toward the center of that circle.
It has a centripetal force acting on it, also directed toward the center of that circle.
"Directed toward the center"="radially".
Any object moving in a circle has a radial acceleration.
That means it has a radial force acting on it.
In addition, the object may be accelerating tangentially , along the direction tangential to the circle or perpendicular to the radius.
Then there is also a tangential force or the net force has a tangential component as well as the radial component.
The total or net acceleration is the vector sum of the radial and tangential components.
The total or net force is the vector sum of the radial and tangential components.
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