Mechanics Resource Packet (comments to:

Suggestions for 'Active' Classroom Learning

The graph at the right describes the positions of masses m1 and m2 in one dimension as a function of the time. Make up a story to say what might have caused this graph..

This graph shows two more tracks,this time of different masses m1 and m2. Again tell the story of what could have caused these graphs. What can you say about the masses m1 and m2 ? What about the masses m1 and m2 in the previous graph?

(Think-share-pair) A little red wagon with a mass of 2 kg is filled with 5 kg of sand. The sand is leaking out of a hole in the wagon at the rate of q kg/s. A motor attached to the wagon is pushing the wagon forward with a thrust of 1 N. The wagon has velocity V at t=0. How would you calculate the acceleration of the cart as a function of the time?

Friction problem (think-share-pair).

A 65-kg skier stands on a horizontal, snowy surface. The coefficient of friction between skis and snow is 0.02. A friend of the skier pushes on her back with a force of 1 N. Calculate the acceleration of the skier. [The acceleration is zero. Students may calculate a negative acceleration.]

Classroom demo. Needs a smooth surface. Comes after impulse has been discussed.

A ruler or meter stick is to be kicked (or flicked) by the instructor. The direction of the kick is in, say, the x-direction. The ruler or meter stick lies along the y-direction and the kick is delivered in the x-direction, but off-center, perhaps at the 35 cm position of the meter stick. The class must predict the place where the meter stick will wind up after being kicked. [Only a few students will predict that the stick will travel in the same direction as the kick is delivered, namely the x-direction. Thanks to Leigh Palmer /phys-L for this suggestion.]

A mass M hangs from a fixed support by a thin thread.An identical thread is connected to the bottom of mass M. By grasping the bottom thread, a person can make either the top thread or the bottom thread break. Discuss how this would be done. [1-2 min, Think-pair-share.Yank sharply on the lower thread to break it first.]

1-Dimensional Motion

[TSP] A crude speed trap consists of one policeman with a whistle who blows it when a car passes him. A second policeman is stationed a distance L away and is armed with a stop watch. If the car reaches the second cop in less than T seconds he flags the car down and issues a speeding ticket. Sketch three possible functions of position versus time, all of which correspond to travelling L in time T. One of the functions should correspond to a car whose speed is actually L/T when it passes the second cop, one whose speed is smaller and one bigger. Is it fair to ticket all three cars?

An advertisement in a popular magazine went as folows: 'Being a student athlete takes more than brains and brawn. It takes time and effort. Endurance and commitment. It's a student athlete who can hang in the air 2.5 seconds while shooting a layup and an athlete who knows what laws of physics keep him there ...'. Comment on the physics underlying this ad. Is there anything unreasonable here?

There are good and bad hang times for punts. Figure out what a 'good' hang time would be and what a 'bad' hang time would be, and give your reasoning (what makes a hang time good or bad?).

Given the following four graphs of x vs. t, say which have positive acceleration, which have negative acceleration, and which have zero acceleration.

Consider the dot product of vectors A1 and A2, the dot product of B1 and B2, and the dot product of C1 with C2.

  a) Which dot product is greatest?
  b) Which doe product is smallest?

The sketch shows six situations where two forces simultaneously act on a box after the box has been sent sliding over a frictionless surface, either to the left or to the right. The forces are either 1 N or 2 N in magnitude, as indicated by the arrow lengths in the figure. For each situation, determine whether the work done on the box by the net force during the displacement d is positive, negative, or zero.

A spring has unstretched length L, and the middle of the spring is marked with a red dot. The spring has force constant k. We extend the spring by a distance x by exerting a force kx on the right-hand end of the spring. Consider the right-hand half of the spring for a moment. It feels the force kx to its right and it must also feel a force to its left at the red dot. a) How big is this force (the force exerted on the righthand half of the spring by the lefthand half of the spring? b) Did the righthand half of the spring extend by an amout x or by a different amount? c) Is the force constant of the righthand half of the spring equal to k or to some other value? d) From the discussion so far would you say that cutting a spring in half would leave the halves with the same spring constant, or a different spring constant (Is half a spring 'stiffer' or 'softer' or the same as the full spring?) [Shorter spring has greater spring constant; half a spring is twice as stiff.]

Springs A and B are identical except that A is 'stiffer' than B (that is, kA > kB). Which spring does more work if they are stretched the same amount? Which spring does more work if they are stretched by the same force? [A does more work when stretched the same. B does more work when force is the same. Very nicely seen on a F vs. x plot. Same stretching, larger slope does more work. Same force, larger slope reaches same force sooner, less area under the curve.]

In picking up a book from the floor and putting it on a table you do work. However, the initial and final velocities are the same, so there is no change in the book's kinetic energy. Does this violate the work-kinetic energy theorem? Explain why or why not.

An earthquake can release enough energy to devastate a city. Where does this energy 'reside' a moment before the earthquake takes place?

Where is the center of mass of the Earth's atmosphere?

A parachutist carrying a pumpkin as a halloween stunt finds that the pumpkin is ripped out of her arms when the chute opens. Explain why, using the notion of impulse.

Comment on the following statement, taken from an examination paper "The collision between two helium atoms is perfectly elastic, so that momentum is conserved."

Lee Iacocca argues that front wheel drive cars are a bad idea because, "there is only so much friction available and you can use it for propulsion or for steering but not both." Do you agree?

A horse refuses to try to pull a cart, saying its impossible beause the cart will pull back with the same force the horse exerts, so the cart will not move. Comment on this line of reasoning. Support your comments with a free-body diagram of the cart.

[In class group activity, up to 5 min.] Two buses are approaching one another.. Each is traveling at 25 km/h, and they are separated by 0.5 km when the driver of one bus tosses a tennis ball straight ahead, at 20 mph with respect to her bus. The bounces elastically off the vertical windshield of the other bus, and back to the driver who threw it. a) What is the relative speed of the ball with respect to the driver when it is caught? [ Neglect gravity.] b) What happens with gravity turned on?

A constant force is applied to a two-block system as shown .
The coefficient of static friction between blocks is 0.25.
a)What is the acceleration of each mass?
b)What is the maximum horizontal force in order that the 5-kg mass remain at rest with respect to the 10-kg mass?
c)What normal force does the 10 kg block exert on the 5-kg block?
The drawing shows an object traveling in a circle. The dots represent the object's position at points equally spaced in time. Draw vectors of appropriate relative length and direction showing the tangential and normal components of the object's acceleration at each point on the diagram.

There is a head-on collision between a 15-kg block and a 5-kg block. The 15-kg block is moving at a speed of 10 m/s and the 5-kg block is moving at a speed of 2 m/s.
During the collision, the average force exerted on the 5-kg block by the 15-kg block is Fon5.
During the collision, the average force exerted on the 15-kg block by the 15-kg block is Fon15.
a) Fon5 is 3 times as great as Fon15
b) Fon5 is more than 3 times as great Fon15
c) Fon5 is less than 3 times as great as Fon15 but more than Fon15
d) Fon5 is equal to Fon15
e) Fon5 is less than Fon15.
f) none of the above

A mass M is suspended from the ceiling by a light string. The mass is set into uniform circular motion as shown in the figure. List all the forces acting on the mass in order of decreasing magnitude. Also indicate the direction of each force.

To create artificial gravity in interstellar space, a donut-shaped space station is rotated about the axis of symmetry. A person throws a ball to a fellow astronaut as shown in the figure. Describe the trajectory of the ball.

(Good classroom problem after mass-spring behavior is discussed, including energy.)

A proposed automobile design has the bumper effectively attached to the body of an 1100-kg car by a spring of constant 150 N/m. The body structure is capable of withstanding a 5.5g's without suffering damage due to crumpling. The car must pass a 'crash test' at 5 mph with a stationary obstacle.
a) Is this a satisfactory design? State the reasons for your conclusion.
[No, g forces are very low, but bumper length is much too great.]
b) If the design is not satisfactory, improve the design to meet the crash test specs without serious objections to your modified design. [The spring constant can be greatly increased and still meet the spec, with a much shorter length of bumper.]

Energy diagrams, forces, and equilibrium (by J. W. Harrell, U of Alabama)

A particle is subject to a conservative force for which the potential energy function U(x) is given by

1. Use Maple to plot U(x) from x= -1 to x= +5 .

2. Examine the plot and estimate the following

3. a) At what points would the particle be in stable equilibrium if placed there at rest?

   b) At what points would the particle be in unstable equilibrium if placed there at rest?

4. Now use Maple to calculate the force function F(x) = -dU(x)/dx.

5. On a single graph, plot both U(x) and F(x). Is your graph consistent with your answers to questions 2 and 3? If not, then reconcile your answers.

6. Now suppose the particle has an energy of 10 units.

On a single graph, plot both U(x) and total energy. (Don't use E for energy; Maple has claimed E for something else.)

7. If no other forces act on the particle, determine from your graph


to differentiate, use diff(f(x),x);

Download maple worksheet

In 'Tale of a Guinea Pig', [p. 174, ff. Bantam, 1981] Geoffrey Page tells of having his spitfire pursued in 1944 by a Me-109 near Lisieux, over France. Page's left leg had become useless due to a cannon shell fragment, and he was at treetop level, jinking, and trying to out-turn the German plane. "The Messerschmitt was just on the edge of a stall when the pilot fired the guns: the recoil slowed the plane sufficiently to flick over and strike the trees twenty feet below." According to Len Deighton, ['Fighter', Ballantine, 1977, p. 91], the Me-109 carried 2 cannon, each of which fired 0.138-kg projectiles at a rate of 520 rounds/minute, with a muzzle velocity of 550 m/s, as well as two machine guns, each of which fired 0.0128-kg projectiles at 1100 rounds/min with a muzzle velocity of 750 m/s.

a) Determine the final speed of the 1800-kg Messerschmitt if its initial speed was 36 m/s, and the pilot fired all four guns for 2 seconds.

b) Determine the average force exerted on the plane by the guns while all four guns were firing.