9.1 impulse and momentum
How is velocity affected by force?
1.Newton’s first Law of motion says if no net force acts on a body its velocity is constant
?What would happen to a rolling ball in the absence of friction?
?How could you move to the shore if you were stuck in the middle of a icy lake with a frictionless surface?
2.Newton’s 2nd Law of motion describes how the velocity of a body is changed by a net force acting on it
Impulse
-product of the average net force exerted on an object and the time interval over which the force exerts and equals and has the unit N s
Linear momentum-product of mass and velocity of an object (p=mv)
?what is an example of linear momentum?
Impulse-momentum theory
p=mv where p = momentum
-the impulse of an object is = to the change in momentum that it causes
Problem:
What is the change in momentum of a tennis ball that was hit by a tennis racket?
-We know that the area under the curve is 1.4N s ?What does this mean?
-the change in the momentum of the ball is 1.4 N s
-remember that N s=kg m/s
?How can we rearrange the impulse-momentum theorem to solve for the momentum of the ball after it was hit?
-so the ball’s final momentum is the sum of the initial momentum and the impulse
-we know that the mass of the ball is .06kg and that the ball was originally at rest
?How can we solve this problem?
-the answer is = to its impulse
?How can we find its velocity?
-large change in momentum occurs only when there is a large impulse
-a large impulse can result either from a large force acting over a short period of time or from a smaller force acting over a longer period of time
?What happens to a driver when a car crash suddenly stops a car?
-an impulse is needed to bring the driver’s momentum to zero
?What in the car can exert a large force during a short period of time?
-a steering wheel works great for this!
?how can an air bag change this?
-an air bag reduces the force exerted on the driver by greatly increasing the length of the time the force is exerted
Problem:
A 2200 kg SUV traveling at 94km/h can be stopped in 21 s by applying the brakes. If the SUV is stopped in .22s after hitting a wall, what is the average force exerted by the SUV in each stop?
Know
m=2200kg
v1=26m/s
v2=0m/s
Change in T=21s and .22s
Unknown
F=?
Formula: F t = p2-p1
Demo-What happens to a person when they jump from a chair onto the floor in regards to their knees?
-think of the impulse-momentum theorem
-how would knee locking vs bending the knees affect the force AND why?
9.2 Conservation of momentum
When is the momentum of the system of two balls conserved?
-closed system-no balls are lost and no balls are gained(don’t knock a pool ball off the table!)
-the forces involved are internal forces meaning that there are no forces acting on the system by objects outside it
Isolated system-when the net external force on a closed system is zero
No system on Earth can be said to be absolutely isolated because there will always be some interactions between a system and its surroundings
Law of conservation of momentum-states that the momentum of any closed, isolated system does not change
Recoil
-the momentum of a baseball changes when the external force of a bat is exerted on it so the baseball is not an isolated system
-after clashing with each other, both skaters are moving, making this situation similar to that of an explosion. Because the push was an internal force, you can use the law of conservation of momentum to find the skater’s relative velocities
-the total momentum of the system was zero before the push therefore, it must be zero after the push
-the coordinate system was chosen so that the positive direction is to the left
-the momenta of the skaters after the push are equal in magnitude but opposite in direction. The backward motion of skater C is an example of recoil.
-the velocities depend on the skaters’ relative masses. The less massive skater moves at the greater velocity.
Two-dimensional collisions
-after the collision, both billiard balls are moving and have momenta
-as long as the friction with the tabletop can be ignored, the system is closed and isolated
-the initial momentum equals the vector sum of the final momenta:
-the equality of the momenta before and after the collision also means that the sum of the components of the vectors before and after the collision must be equal
-if the x-axis in the direction of the initial momentum then the y-component of the initial momentum is equal to zero
-the sum of the final y-component must be zero
-the y-components are equal in magnitude but are in the opposite direction and have opposite signs. The sum of the horizontal components also is equal.
Conservation of angular momentum
-states that if no net external torque acts on an object, then its angular momentum does not change
-an object’s initial angular momentum is equal to its final angular momentum
-Earth spins on its axis with no external torque so its angular momentum is constant and conserved so the length of a day does not change
-if a torque-free object starts with no angular momentum, it must continue to have no angular momentum
-if part of an object rotates in one direction, another part must rotate in the opposite direction
Example-if you switch on a loosely held electric drill, the drill body will rotate in the direction opposite to the rotation of the motor and bit
-because of the conservation of the angular momentum, the direction of rotation of a spinning object can be changed only by applying a torque
-when a top is vertical, there is no torque on it and the direction of its rotation does not change
-if the top is tipped, torque tries to rotate it downward. Rather than tipping over, the upper end of the top revolves or precesses slowly about the vertical axis.
-a gyroscope is a wheel or disk that spins rapidly around one axis while being free to rotate around one or two other axes
-the direction of its large angular momentum can be changed only by applying an appropriate torque. Without such a torque, the direction of the axis of rotation does not change.
-gyroscopes are used in airplanes, submarines, and space crafts to keep unchanging reference in direction
9.2 conservation of momentum
Two particle collisions
-during the collisions of two balls, each briefly exerts a force on the other
-the forces that they exert on each other are equal and opposite in reaction
?How do the impulses compare?
-they are equal in magnitude and opposite in direction
?How can we find the momentum?
*F t=p2-p1
Ball 1 pA2=FBonA t + pA1
Ball 2 pB2=FAonB t + pB1
Conservation of momentum
pA2 + pB2 = pA1 + pB1
?what does this mean?
-the sum of the momentum of the balls is the same before and after the collision
Momentum in a closed system
Closed system-a system that does not gain or lose mass
Internal forces-all the forces within a closed system
Isolated system-when the net external force on a closed system is zero
Law of conservation of momentum
States that the momentum of any closed system with no net external force does not change
Problem: car collisions
A 2275 kg car is going 28m/s and rear ends a 875 kg compact car on ice going 16m/s in the same direction. The two cars stick together. How fast does the wreckage move immediately after the collision?
mA = 2275 kg
vA1 = 28m/s
mB = 875 kg
vB1 = 16 m/s
v2=unknown
remember that momentum = impulse
remember that F t = m v
?what are some calculations to use?
Conservation of momentum
p1=p2
pA1 + pB1 = pA2 + pB2
mAvA1 + mBvB1 = mAvA2 + mBvB2
velocities are equal
vA2=vB2=v2
mAvA1 +mBvB1=(mA + mB)v2
v2= mAvA1 +mBvB1
mA + mB
v2=(2275kg)(28m/s) + (875kg)(16m/s)
2275 kg + 875kg
V2=25m/s
Explosions
Example
Skater A gives skater B a “push”
with the total momentum being 0 before and after the push. The momentum of the skaters after the push are equal in magnitude BUT opposite in direction. The backward motion of the skater after the push is an example of recoil.
?How does a rocket in space change its velocity?
-after chemicals are mixed producing hot gases that leave the exhaust nozzle at high speeds
-the law of conservation of momentum can be applied
Problem
An astronaut at rest in space fires a thruster pistol that expels 35g of hot gas at 875m/s. The combined mass of the astronaut and pistol is 84kg. How fast and in what direction is the astronaut moving after firing the pistol?
(look at picture p212)
mA=84kg
mB=.035kg
vA1 = vB1=0m/s
vB2=-875m/s
vA2=?
Calculations
p1=pA1+pB1=0
pA1 + pA2 = pA2 + pB2
0 =pA2 + pB2
pA2 = -pB2
mAvA2=-mBvB2
so solve for vA2=-(mBvB2)
mA
vA2=-(.035kg)(-875m/s) = +.36m/s
84kg
Angular Momentum
-the quantity of motion used with objects rotating around a fixed axis
-changes when torque acts on an object
-(review) torque-product of the applied force and the lever arm which is the perpendicular distance from the axis of rotation to a line along which the force acts
-rotational inertia-resistance to change in angular velocity
comparison
comparison
p=mv to L=Iw
w=angular velocity
I=rotational inertia
L=angular momentum
Variations in these equations
Trends
-if the torque is 0 then change in L is 0
-no change in angular momentum over a period of time it must be conserved
-if an object is rotating at a given velocity about a fixed axis, its rotational inertia must increase if its angular velocity decreases and vice versa