# Chapter 15

```15.1 Objective
1. Calculate the linear momentum of a particle and linear impulse of a force.
2. Apply the principle of linear impulse and momentum.
Applications
A good example of impulse is the
action of hitting a ball with a bat.
The impulse is the average force
exerted by the bat multiplied by the
time the bat and ball are in contact.
Is the impulse a vector? Is the impulse pointing in the same direction as the
force being applied?
Given the situation of hitting a ball, how can we predict the resultant
motion of the ball?
Continue…
When a stake is struck by a
sledgehammer, a large impulse
force is delivered to the stake and
drives it into the ground.
If we know the initial speed of the
sledgehammer and the duration of
impact, how can we determine the
magnitude of the impulsive force
delivered to the stake?
A hard lesson
Another one
Linear impulse
Principle of linear impulse and momentum
Example
• A golf ball having a mass of 40 g is struck with a force profile F = 200
sin(100t). Find a) the ball’s velocity leaving the tee; b) the distance the ball
will travel (assume flat range)
Example
• Force on a 2g bullet, as it travels horizontally through the barrel, varies as
shown. Neglect friction. Find: F0 (maximum net force) if the bullet velocity is
500m/s at t=0.75 ms.
Example
• A car traveling at 4 ft/sec (2.72 mph) weighing 2700 lb crashes into a wall.
The duration of the impact is 0.06s.
• Find: a) average impulsive force during collision if brakes are not applied
• B) average impulsive force if all tires brake (k=0.3)
15.2-15.3 objective
1. Apply the principle of linear impulse and momentum to a system of
particles.
2. Understand the conditions for conservation of momentum.
Applications
This large crane-mounted hammer is used to drive
piles into the ground.
Conservation of momentum can be used to find the
velocity of the pile just after impact, assuming the
hammer does not rebound off the pile.
If the hammer rebounds, does the pile velocity change from
the case when the hammer doesn’t rebound ? Why ?
In the impulse-momentum analysis, do we have to consider
the impulses of the weights of the hammer and pile and the
resistance force ? Why or why not ?
Example
• A train has one engine (50 Mg) and 3 cars (30 Mg each). It takes 80 seconds
for the train to uniformly increase speed to 40 km/hr, starting from rest.
• Find: a) the coupling force between the engine and first car,
• B) the traction force of the engine (Assume all cars roll freely)
https://lancastercatholichs.instructure.com/courses/410/files/16
Non-impulsive forces and impulsive forces
example
• A barge B weighs 30,000 lb and supports a 3,000 lb car C (the barge is not
secured to the pier). The car is driven 200 ft across the barge. Neglect water
resistance.
• Find: how far the barge moves from the pier
Example
• A 5 kg spring-loaded gun rests on a smooth surface. It fires a 1 kg ball with a
velocity of 6m/s relative to the gun as shown.
• Find: the separation distance d between ball and gun.
Example
• A barge weighs 45,000 lb and supports two cars A and B, weighing 4,000 lb
and 3,000 lb, respectively. They start from rest and accelerate towards each
other (aA = 4ft/s2, aB = 8 ft/s2) until they reach constant speeds of 6 ft/s
(relatively to the barge). Initially, the barge is at rest. Neglect water resistance.
• Find: the speed of the barge just before impact
15.4 Objective
1. Understand and analyze the mechanics of impact.
2. Analyze the motion of bodies undergoing a collision, in both central and
oblique cases of impact.
Collision
Applications
The quality of a tennis ball is measured by the height of its
bounce. This can be quantified by the coefficient of
restitution of the ball.
If the height from which the ball is dropped and the height of
its resulting bounce are known, how can we determine the
coefficient of restitution of the ball?
Continue…
In the game of billiards, it is important to be able to predict
the trajectory and speed of a ball after it is struck by
another ball.
If we know the velocity of ball A before the impact, how can
we determine the magnitude and direction of the velocity of
ball B after the impact?
What parameters would we need to know to do this?
Central and oblique impact
Phases of central impact
Example
• Disk A (2kg) slides on a smooth surface (VA1 = 5m/s) and strikes Disk B
(4kg, VB1=2m/s) with central impact (e=0.4).
• Find: VA2 and VB2
Example
• A 2kg ball strikes a suspended 20 kg block with a velocity of 4m/s. The
coefficient of restitution is 0.8.
• Find: the height h to which the block will swing before it momentarily stops.
Oblique impact
Example
• A girl throws a ball with horizontal velocity v1=8ft/s. The “e” between the
ball and the ground is 0.8.
• Find: a) the velocity after the ball rebounds, and
• B) the maximum height the ball rises after the first bounce.
15.5-15.7 Objective
1. Determine the angular momentum of a particle and apply the principle of
angular impulse & momentum.
2. Use conservation of angular momentum to solve problems.
Applications
Planets and most satellites move in elliptical orbits. This
motion is caused by gravitational attraction forces. Since
these forces act in pairs, the sum of the moments of the forces
acting on the system will be zero. This means that angular
momentum is conserved.
If the angular momentum is constant, does it mean the linear
momentum is also constant? Why or why not?
Applications
The passengers on the amusement-park ride
experience conservation of angular momentum
about the axis of rotation (the z-axis). As shown
on the free body diagram, the line of action of
the normal force, N, passes through the z-axis
and the weight’s line of action is parallel to it.
Therefore, the sum of moments of these two
forces about the z-axis is zero.
If the passenger moves away from the z-axis,
will his speed increase or decrease? Why?
Angular momentum
Direction
Example
• Determine 0 for the 1.5 kg particle below:
Newton’s 2nd law
Example
• Two spheres, each 3 kg, are attached to a rod of negligible mass. A torque M
= 6e0.2t (N/m) is applied as shown, starting from rest.
• Find: speed of spheres after 2 seconds
Conservation of angular momentum
An example of this condition occurs when a particle is subjected
only to a central force. In the figure, the force F is always directed
toward point O. Thus, the angular impulse of F about O is always
zero, and angular momentum of the particle about O is conserved.
Example
• An amusement park ride consists of a car attached by cable to point O. It
rotates in a horizontal plane, v1 = 4ft/s where r = 12 ft. The cable is then
retracted at a constant rate of 0.5 ft/s.
• Find: the speed of the car after 3 seconds.
Example
• A 0.1 kg block is given a horizontal velocity v1 = 0.4 m/s when r1 = 500 mm.
It slides along a smooth conical surface.
• Find: the block’s speed and angle  when h = 100mm.
```