Mechanics

One exciting thing from mechanics is the drifting of a bicycle on a slippery surface with loose sediments.

Quantities in Mechanics

Name	Symbol	Unit	Type
average speed	v	m/s, meters/second	scalar
distance	d	m, meter	scalar
time	t	s, second	scalar
average velocity	v	m/s, meters/second	vector
displacement	Δx	m, meter	vector
velocity, instantaneous velocity	v	m/s, meters/second	vector
acceleration	a	m/s², meters per second²	vector
Force	F	N, newtons	vector
mass	m	kg, kilogram	scalar
Coefficient of friction	μ	No unit!	scalar
Momentum	p	kg · m/s, kilograms · m/s	vector
Impulse	J	N · s, newtons · second	vector
Power	P	W, watts	scalar
centripetal acceleration	a_c	m/s², meters per second²	vector
centripetal force	F_c	N, newtons	vector
radius	r	m, meters	scalar

Key Things to Know in Mechanics

Inertia and mass are the same thing. Inertia or mass is a measure of an object's resistance to change in motion.
Constant velocity and zero velocity means the net force is zero and acceleration is zero.
Weight is force due to gravity = F_g = mg, while mass = m
Weight (force of gravity) decreases as you move away from the earth by distance squared.
Velocity, displacement [s], momentum, force (weight), torque, and acceleration are vectors.
Speed, distance [d], time, length, mass, temperature, charge, power and energy (joules) are scalar quantities.
The slope of the distance-time graph is velocity.
The slope of the velocity-time graph is acceleration.
The area under a velocity-time graph is distance.
Magnitude is a term used to state how large a vector quantity is.
At 0° two vectors have a resultant equal to their sum.
At 180° two vectors have a resultant equal to their difference.
From the difference to the sum is the total range of possible resultants.
Centripetal force and centripetal acceleration vectors are toward the center of the circle,
while the velocity vector is tangent to the circle.
An unbalanced force (object not in equilibrium) must produce acceleration.
If an object is not accelerating, it is in equilibrium and no unbalanced forces are acting.
The equilibrant force is equal in magnitude but opposite in direction to the resultant vector.
Momentum is conserved in all collision systems.

Key Words in Mechanics

acceleration, position, displacement, distance, normal force, inertia, momentum

Distance: the length traveled by an object.
Displacement: the change of an object's position.
Position: an object's location from the chosen origin.
Acceleration: the rate of change of an object's velocity.
Normal force: perpendicular to the surface.
inertia: the mass. Mass is a measure of an object's resistance to change in motion.
Momentum: the product of the mass of an object and its velocity.

Formula in Mechanics

v = \( \frac{d}{t} \)
a = \( \frac{Δv}{t} \)
v_f = v_i + a t
d = v_i t + \( \frac{1}{2} \) a t²
v_f² = v_i² + 2ad
F_net = m a
F_f = μ F_N
F_g = \( \frac{G m_1 m_2}{r^2} \)
g = \( \frac{F_g}{m} \) or F_g = mg
p = m v
p_before = p_after
m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f
J = F_nett
F_nett = Δp
F_s = k x
F_c = m a_c
a_c = \( \frac{v^2}{r} \)
P = \( \frac{W}{t} \)

Example 1

A car traveling on a straight road at 15.0 meters per second accelerates uniformly to a speed of 21.0 meters per second in 12.0 seconds. The total distance traveled by the car in this 12.0-second time interval is
(1) 36.0 m (3) 216 m
(2) 180. m (4) 252 m

List the information: v_i = 15 m/s; v_f = 21 m/s; t = 12 s; d = ?
Write out the equation: d = v_i t +

\frac{1}{2}

a t² Substitute numbers with unit: d = 15 m/s × 12 s + 0.5 a (12 s)²
But wait! what is a? Find a: a = \( \frac{v_f - v_i}{t} = \frac{21 m/s - 15 m/s}{12 s} \) = 6 m/s ÷ 12 s = 0.5 m/s²
d = 15 m/s × 12 s + 0.5 × 0.5 m/s² × (12 s)²
= 216 m

Example 2

A 0.149-kilogram baseball, initially moving at 15 meters per second, is brought to rest in 0.040 second by a baseball glove on a catcher’s hand. The magnitude of the average force exerted on the ball by the glove is
(1) 2.2 N (3) 17 N
(2) 2.9 N (4) 56 N

List the information: m = 0.149 kg; v_i = 15 m/s; v_f = 0 m/s; t = 0.04 s; F_net = ?
Write out the equation: F_net = \( \frac{Δp}{t}\)
Substitute numbers with unit: F_net = \( \frac{mv_f - mv_i}{t} \)
= (0.149 kg × 0 m/s - 0.149 kg × 15 m/s) ÷ 0.04 s = 55.9 N

Example 3

A student pulls a 60.-newton sled with a force having a magnitude of 20. newtons. What is the magnitude of the force that the sled exerts on the student?
(1) 20. N (3) 60. N
(2) 40. N (4) 80. N

A student pulls on a sled with a force of 20 N forward. By Newton's third law, the sled pulls back on the student with a force of 20 N.

Example 4

A go-cart travels around a flat, horizontal, circular track with a radius of 25 meters. The mass of the go-cart with the rider is 200. kilograms. The magnitude of the maximum centripetal force exerted by the track on the go-cart is 1200. newtons.

a) What is the maximum speed the 200.-kilogram go-cart can travel without sliding off the track?
(1) 8.0 m/s (3) 150 m/s
(2) 12 m/s (4) 170 m/s

r = 25 m; m = 200 kg; F_c = 1200 N; v = ?
F_c = \( \frac{mv^2}{r} \)
v = √(F_c r ÷ m) = √(1200 N × 25 m ÷ 200 kg) = 150 m/s

b) Which change would increase the maximum speed at which the go-cart could travel without sliding off this track?
(1) Decrease the coefficient of friction between the go-cart and the track.
(2) Decrease the radius of the track.
(3) Increase the radius of the track.
(4) Increase the mass of the go-cart.

v = √(F_c r ÷ m). Choices 2 and 4 decrease v. Choice 1 decreases F_c, which also decreases v. Choice 3 is correct because a larger r inreases v.

Exercises

1. Which body is in equilibrium?
(1) a satellite moving around Earth in a circular orbit
(2) a cart rolling down a frictionless incline
(3) an apple falling freely toward the surface of Earth
(4) a block sliding at constant velocity across a tabletop

2. A 3.1-kilogram gun initially at rest is free to move. When a 0.015-kilogram bullet leaves the gun with a speed of 500. meters per second, what is the speed of the gun?
(1) 0.0 m/s (3) 7.5 m/s
(2) 2.4 m/s (4) 500. m/s

3. A ball is thrown vertically upward with an initial velocity of 29.4 meters per second. What is the maximum height reached by the ball? [Neglect friction.]
(1) 14.7 m (3) 44.1 m
(2) 29.4 m (4) 88.1 m

4. On a snow-covered road, a car with a mass of 1.1 × 10³ kilograms collides head-on with a van having a mass of 2.5 × 10³ kilograms traveling at 8.0 meters per second. As a result of the collision, the vehicles lock together and immediately come to rest. Calculate the speed of the car immediately before the collision. [Neglect friction.] [Show all work, including the equation and substitution with units.]

5. What is the weight of a 2.00-kilogram object on the surface of Earth?
(1) 4.91 N (3) 9.81 N
(2) 2.00 N (4) 19.6 N

6. When a 12 N horizontal force is applied to a box on a horizontal tabletop, the box remains at rest. The force of static friction acting on the box is
(1) 0 N (3) between 0 N and 12 N
(2) 12 N (4) greater than 12 N