Chapter 05 : Using Newton's Laws

Review : Test 2 Practice problem HW04-15

A 14 kg bucket is being lowered vertically by a rope in which there is 132 N of tension.
(a) What is the acceleration of the bucket? (Up or down?)
Suppose the mass of the bucket is 12 kg and the tension is the same 132 N.
(b) What is the acceleration of the bucket? (Up or down?)

(Sketch what v(t) looks like in each case.)
Remember: velocity and acceleration (a=dv/dt) are two completely different things.
The sign of one tells you nothing about the sign of the other.

Review : Test 2 Practice problem HW05-YY

A 10 kg box is sliding down a rough 30^o incline as shown in the figure. In the upper position, the box is sliding down the incline at 50 cm/s but we observe that the box is slowing down and comes to a stop after sliding 50 cm along the ramp.
(a) What must the kinetic coefficient of friction be?

Review : Test 2 Practice problem HW05-ZZ

A box of unknown mass M is sliding across the floor to the right at 1.5 m/s and is being pushed with a constant force of F_push=200 N.
The (flat, horizontal) floor is rough, with a coefficient of kinetic friction of μ_k=0.6 and we observe that the box is slowing down, coming to a stop after travelling 50 cm.
(a) What must the mass of the box be?

Jurassic Park 2 : Version 2

At one point in the movie, a dinosaur has pushed a trailer of mass 2000 kg over the edge of a cliff. It hangs there, connected by a cable to another trailer of mass 4000 kg which remains on the ground as shown in the figure.

We observe that nothing is moving here.
What must the coefficient of STATIC friction be?
We observe that the trailers are moving at a constant speed (the 4000 kg object sliding to the right, and the 2000 kg object moving downward).
What must the coefficient of KINETIC friction be?

(Ignore the μ_s value in the figure.)

Jurassic Park 2 : Version 3

At one point in the movie, a dinosaur has pushed a trailer of mass 2000 kg over the edge of a cliff.
It hangs there, connected by a cable to another trailer of mass 4000 kg which remains on the ground as shown in the figure.
Let the coefficients of friction here be μ_s=0.8 and μ_k=0.6

(a) Verify that the present situation is stable (i.e. static friction is providing enough force to keep the trailers from sliding over the edge).

(b) The dinosaur is not happy with that and decides to push horizontally on the trailer. How much force does the dinosaur need to provide to overcome static friction and cause the trailers to start to move?

(c) If the dinosaur continues to push with the same amount of force, determine the acceleration present, and the tension in the connecting cable.

Uniform Circular Motion

Suppose we have an object moving along a circular path such that it's speed remains the same value.

Velocity is a vector and the velocity vector keeps changing direction as the object moves around the circle, so the only constraint we've added is that it's speed (i.e. the magnitude of the velocity: |v|) is constant.

An object moving in a circle at a constant speed is said to be undergoing UNIFORM CIRCULAR MOTION.

Some Useful Definitions

T : PERIOD : the time per rotation (or cycle or revolution or ...)
f : FREQUENCY : rotations per time, so f=1/T

UNITS: ideally rotations per second: s^-1 , also called Hertz (Hz)
Commonly encountered unit: RPM (revolutions per minute)

Constant speed, so object is travelling around a circle of radius r:

v=(2πr)/T = 2πrf (travels one circumference in one period)
T = 2πr/v

Radial Acceleration

Here we will show that acceleration is present in circular motion, even though the speed of the object is constant.

Result: a radially inward acceleration of magnitude:
\[ a_r = \frac{v^2}{r} \]

Example: CD-ROM (DVD, Blu-ray, ...)

Estimate the radial acceleration at the outer edge of a CDROM that is spinning in a 40X drive.
Note: original 1X drives spun at approximately 360 RPM,
40X means it spins the disk at 40 times that rate or 14,400 revolutions/minute.

What is f here? (In proper units!)

The diameter of the disk is 120 mm (part of the standard).

How fast is a point on the outer edge of the CD moving?
What is the (radial) acceleration of that point?
(In m/s² and in g's.)

Polycarbonate plastic: very long chains

High enough RPMs and...

CD shattering video (see just after the 6 minute mark)

Review : Test 2 Practice problem HW04-15
A 14 kg bucket is being lowered vertically by a rope in which there is 132 N of tension. (a) What is the acceleration of the bucket? (Up or down?) Suppose the mass of the bucket is 12 kg and the tension is the same 132 N. (b) What is the acceleration of the bucket? (Up or down?)
(Sketch what v(t) looks like in each case.) Remember: velocity and acceleration (a=dv/dt) are two completely different things. The sign of one tells you nothing about the sign of the other.
Review : Test 2 Practice problem HW05-YY
A 10 kg box is sliding down a rough 30^o incline as shown in the figure. In the upper position, the box is sliding down the incline at 50 cm/s but we observe that the box is slowing down and comes to a stop after sliding 50 cm along the ramp. (a) What must the kinetic coefficient of friction be?
Review : Test 2 Practice problem HW05-ZZ
A box of unknown mass M is sliding across the floor to the right at 1.5 m/s and is being pushed with a constant force of F_push=200 N. The (flat, horizontal) floor is rough, with a coefficient of kinetic friction of μ_k=0.6 and we observe that the box is slowing down, coming to a stop after travelling 50 cm. (a) What must the mass of the box be?

Uniform Circular Motion
Suppose we have an object moving along a circular path such that it's speed remains the same value. Velocity is a vector and the velocity vector keeps changing direction as the object moves around the circle, so the only constraint we've added is that it's speed (i.e. the magnitude of the velocity: \|v\|) is constant. An object moving in a circle at a constant speed is said to be undergoing UNIFORM CIRCULAR MOTION.
Some Useful Definitions
T : PERIOD : the time per rotation (or cycle or revolution or ...) f : FREQUENCY : rotations per time, so f=1/T UNITS: ideally rotations per second: s^-1 , also called Hertz (Hz) Commonly encountered unit: RPM (revolutions per minute) Constant speed, so object is travelling around a circle of radius r: v=(2πr)/T = 2πrf (travels one circumference in one period) T = 2πr/v

Radial Acceleration
Here we will show that acceleration is present in circular motion, even though the speed of the object is constant.

Result: a radially inward acceleration of magnitude: \[ a_r = \frac{v^2}{r} \]

Example: CD-ROM (DVD, Blu-ray, ...)
Estimate the radial acceleration at the outer edge of a CDROM that is spinning in a 40X drive. Note: original 1X drives spun at approximately 360 RPM, 40X means it spins the disk at 40 times that rate or 14,400 revolutions/minute. What is f here? (In proper units!) The diameter of the disk is 120 mm (part of the standard). How fast is a point on the outer edge of the CD moving? What is the (radial) acceleration of that point? (In m/s² and in g's.)
Polycarbonate plastic: very long chains	High enough RPMs and...
CD shattering video (see just after the 6 minute mark)