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Physics 221, February 16
Key Concepts:
•Vectors
•Projectile motion
•Friction
•Uniform circular motion
Vector addition
Add two vectors, A an B.
R=A+B
Find the x- and y-components of each vector.
Find the components of R.
Find the magnitude and direction of R.
What is the magnitude an direction of the vector C = A + B?
1.
2.
3.
4.
5.
4,
2,
8,
1.41,
2.83,
45 deg
270 deg
225 deg
45 deg
225 deg
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Extra credit:
Vector A has magnitude 10 and vector B has magnitude 15. What is NOT a
possible magnitude of vector C = A + B?
1.
2.
3.
4.
5.
4
8
12
16
20
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F = ma: no acceleration  no net force
If the component of the acceleration in a certain direction is
zero, then the component of the net force in that direction is
zero.
On the incline the mass does not accelerate in a direction
perpendicular to the incline.
The component of the net force perpendicular
to the incline is zero.
Normal force: n = mg cosθ
Along the incline: ma = mg sinθ
A 30 kg penguin slides down the side of a glacier that has a constant slope of
50°. What is the magnitude of the acceleration of the penguin and what is
the magnitude of the normal force it feels? Neglect any friction.
1.
2.
3.
4.
a = 7.5 m/s2, n = 189 N
a = 9.8 m/s2, n = 294 N
a = 6.3 m/s2, n = 225 N
a = 9.8 m/s2, n = 300 N
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Two blocks, one sitting on a table and the other heavier one hanging over its
edge, are connected by a light string as shown in the figure. Which force
makes the block on the table move, the tension in the string or the weight of
the hanging block?
1. the tension
2. the weight
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What is the acceleration of the blocks?
The blocks accelerate together, the acceleration a
is the same for both blocks.
(1) m1g – T = m1a
(2) T = m2a
Insert (2) into (1).
m1g – m2a = m1a
Solve for a.
a = m1g/(m1+m2)
Friction
The frictional force always acts between two surfaces, and opposes the
relative motion of the two surfaces.
Static friction:
Kinetic friction:
fs ≤ μsN
fk = μkN
N = magnitude of the force pressing the surfaces together
What is the direction of the frictional force on the block if
i)
ii)
iii)
the block is at rest on the ramp?
the block moves up the ramp?
the block is moves down the ramp?
Assume the car is accelerating towards the right. A cup of coffee left on the
roof is accelerating with the car at the same rate. What is the direction of the
frictional force on the cup?
1.
2.
3.
4.
5.
Towards the right
Towards the left
Up
Down
Down and towards
the left
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When sliding a heavy box across the floor, it is often times helpful to lift up on
it slightly because
1.
2.
3.
4.
you are lessening the friction force
by reducing the normal force the
floor exerts on the box.
you are reducing the weight of the
box.
you are working against friction by
lifting up.
you are working against gravity,
even if the box does not move
upwards.
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Extra credit:
You are walking with your friend along a beautiful shoreline and begin talking about
physics. She says “I completely understand friction from physics class! It always acts
against the motion of an object, slowing it down. Your friend’s statement is correct.
1. Yes
2. No
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Projectile motion
Projectile motion is motion of a particle through a region of
space where it is subject to constant acceleration.
Let the acceleration be along the y-direction and let the
trajectory lie in the xy-plane.
Then vx = v0x, x = v0xt, vy= v0y + at, y = v0yt + (1/2)at2.
The motion along the x-direction is independent of the motion
along the y-direction.
If a = -g then
vx = v0x, x = v0xt,
vy= v0y - gt, y = v0yt - (1/2)gt2.
A ball is thrown horizontally from the roof of a 25 m tall building with a
speed of 20 m/s. What is its acceleration just before it hits the
ground?
1.
2.
3.
4.
0
25 m/s2, horizontal
9.8 m/s2, downward
greater than 9.8 m/s2 to
the right and downward.
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Three identical balls are thrown off a building, all with the same initial
speed. Ball 1 is thrown horizontally, ball 2 is thrown at some angle
above the horizontal, and ball 3 is thrown at some angle below the
horizontal. Rank the speeds of the balls as they reach the ground.
1.
2.
3.
4.
v1 = v 2 = v 3
v3 > v 1 > v 2
v1 > v 2 = v 3
v2 > v 1 > v 3
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Demo
• Ballistic cart
http://www.youtube.com/watch?v=MhlTbNYh7VM
Ball and cart have the same constant vx. The ball’s and the
carts horizontal motions are the same. The ball’s vertical
motion is independent of its horizontal motion and does not
influence its horizontal motion.
• Shoot the target
http://www.youtube.com/watch?v=7y_ncpf81k8
Without gravity, the arrow would hit the original position of
the target. With gravity, arrow and target accelerate
downward at the same rate.
Circular motion
An object moving in a circle of radius r with speed v is
accelerating.
This acceleration is called radial acceleration or centripetal
acceleration.
This acceleration, ac, points towards the center of the circle.
The magnitude of the centripetal acceleration vector is
ac = v2/r.
A force is required to make an object move in a circle.
This force is called the centripetal force, with magnitude
Fc = m v2/r.
Fc points towards the center of the circle.
Apparent weight
When you stand on a bathroom scale in an
inertial frame, such as this room, its reading
Is proportional to your real weight.
When you stand on a bathroom scale
in an accelerating frame, such as an
elevator accelerating upward, its reading is
proportional to your apparent weight.
In every accelerating frame we have
wapparent = wreal - ma.
For the elevator accelerating upward:
wapparent = wreal – ma
The apparent weight of a mass m is
its real weight minus its mass times
the acceleration of the frame
(vector addition).
In outer space, where wreal = 0,
wapparent = – ma
Some engineers have suggested that we can simulate gravity in outer space by having
a circular rotating space station where persons feel an outward-directed fictitious
force due to the rotation of the station. The reason they feel such a force is because
1.
2.
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their velocity is toward the center of
the space station and their inertia
tends to keep them moving
outward.
they are accelerating toward the
center of the space and the walls of
the space station provide the
centripetal force, which they
experience as an apparent weight
their velocity is away from the
center of the space station and their
inertia tends to make them move in
towards the center.
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Extra credit:
A ring shaped space station with a radius of 1 km is spinning, so that the
speed of the rim is 50 m/s. A 60 kg man sits on the inside of the rim. What is
his apparent weight?
1.
2.
3.
4.
5.
150000 N
3000 N
588 N
150 N
3N
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