This week's overview covers forces in two dimensions, inclined planes, circular motion, and rotation. Specifically: - Forces in two dimensions are examined, showing how to find the resultant acceleration when two forces act at right angles to each other on an object. - Uniform circular motion and centripetal acceleration are discussed, along with examples of banking and centripetal force. - Inclined planes and the forces that act on objects moving up or down a plane are analyzed. - Projectile motion, where an object moves in a parabolic path due to both horizontal and vertical forces, is introduced. - Rotational motion, including angular displacement, velocity, acceleration, torque, rotational inertia, and

1. Week 3 Overview
Last week, we covered multiple forces acting on an object. This
week we will cover motion in two dimensions, inclined planes,
circular motion, and rotation.
Forces in Two Dimensions (1 of 2)
So far you have dealt with single forces acting on a body or
more than two forces that act parallel to each other. But in real
life situations more than one force may act on a body. How are
Newton's laws applied to such cases? We will restrict the forces
to two dimensions.
Since force and acceleration are vectors, Newton's law can be
applied independently to the X and Y-axes of a coordinate
system. For a given problem you can choose a suitable
coordinate system. But once a coordinate system is chosen, we
have to stick with it for that problem. The example that follows
shows how to find the acceleration of a body when two forces
act on it at right angles to each other.
Forces in Two Dimensions (2 of 2)
To find the resultant acceleration we draw an arrow OA of
length 3 units along the X-axis and then an arrow AB of length
4 units along the Y-axis. The resultant acceleration is the arrow
OB with the length of 5 units. Therefore, the acceleration is 5
m/s2 in the direction of OB. Also when you measure the angle
AOB with a protractor, we find it to be 53°.
The acceleration caused by the two forces is 5 m/s2 at an angle
of 53°.
Uniform Circular Motion

2. When an object travels in a circular path at a constant speed, its
motion is referred to as uniform circular motion, and the object
is accelerated towards the center of the circle. If the radius of
the circular path is r, the magnitude of this acceleration is ac =
v2 / r, where v is its speed and ac is called the centripetal
acceleration. A centripetal force is responsible for the
centripetal acceleration, which constantly pulls the object
towards the center of the circular path. There cannot be any
circular motion without a centripetal force.
Banking
When there is a sharp turn in the road or when a turn has to be
taken at a high speed as in a racetrack, the outer part of the road
or the track is raised from the inner part of the track. This is
called banking. It provides additional centripetal force to a
turning vehicle so that it doesn't skid.
The angle of banking is kept just right so that it provides all the
centripetal force required and a motorist does not have to
depend on the friction force at all.
Inclined Planes
Forces on an Inclined Plane
The inclined plane is a device that reduces the force needed to
lift objects. Consider the forces acting on a block on an inclined
surface. The inclined surface exerts a normal force FN on the
block that is perpendicular to the incline. The force of gravity,
FG, points downward. If there is no friction, the net force, Fnet,
acting on the block is the resultant of FN and FG. By Newton's
second law the net force must point down the incline because
the block moves only along the incline and not perpendicular to
it.
The vector triangle shows that the magnitude of the net force is
always less than the weight, FG.

3. Example:
A 2 kg block is on an incline which is only able to support 16 N
of its weight. Find the acceleration of the block along the
incline.
Solution
:
First, from FG = mg, a 2 kg block weighs 20 N. Also, the
normal force is given as 16 N. We now draw a free body
diagram showing all the forces acting on the block:
We can find the net force on the block using Pythagorean’s
Theorem:
Now, we can put Fnet into F=ma to find the acceleration of the
block:
The block will accelerate down the incline at 6 m/s2.

4. Projectile Motion (1 of 2)
When a ball is thrown or a shell is fired from a gun at an angle
to the horizontal, the ball or the shell follows a curved path
known as a parabola. The motion is in two dimensions and is
called projectile motion. The moving object is called a
projectile.
The vertical motion and horizontal motion can be analyzed
separately for a projectile. The horizontal direction can be the
x-axis and the vertical direction the y-axis.
Vertical Motion
In the vertical direction only the force of gravity acts on the
projectile. The vertical motion of the projectile is the same as
the motion of an object thrown vertically upward with the same
initial vertical velocity.
If vy is the initial vertical component of velocity, the projectile
will reach a maximum height of h = vy2 / (2 g), which is the
maximum height reached by an object thrown vertically upward
with velocity vy.
The time for which the projectile remains in flight is again
determined by the initial vertical component of velocity and is
given by t = 2 vy / g, which is also the total time for which an

5. object thrown vertically upward with velocity vy stays in the
air.
Using the navigation on the left, please proceed to the next
page.
Projectile Motion (2 of 2)
Horizontal Motion
In the horizontal direction there is no force acting on the
projectile. So by Newton's law there is no acceleration in the
horizontal direction. If the projectile is given an initial velocity
of vx it remains the same throughout the duration of the flight.
The horizontal distance, x, that the projectile travels is given
by:
x = vxt
The total time of flight has already been stated to be 2 vy / g.
Therefore, the range is:
x = vx (2 vy / g)
x = 2 vx vy / g.
The range is determined by both the horizontal and vertical

6. components of velocity.
Rotational Motion (1 of 2)
The CD drive in your computer makes the CD spin at a high rate
to either read or write data. This is an example of rotation about
a fixed axis. Until now, we have dealt with objects that translate
or move along a straight or curved path.
Just as the motion along a path is described in terms of
distance, speed, and acceleration, rotational motion is measured
in terms of the angular position, speed, and acceleration of a
body.
Angular Displacement, Velocity, and Acceleration
Angular displacement is the angle through which a body rotates
and is measured in radians. When a body makes one full
rotation its angular displacement, θ, is 360° or 2π radians or
6.28 radians. A radian is approximately 57.3°. You should
always use radians when working with any circular motion
problem. To convert between units, one revolution = 360
degrees = 2π radians.
To convert an angular displacement to a distance, multiply by
the radius of the circle:

7. A rotating rigid body has an angular speed. For translation,
speed is the rate at which distance is covered. Angular velocity,
ω, is the rate at which the angular position of the body changes.
To convert from an angular velocity to a tangential velocity,
multiply by the radius of the circle:
If the angular velocity of a body changes, the body has an
angular acceleration, α. Angular acceleration is the rate at
which angular velocity changes.
To convert from an angular acceleration to a tangential
acceleration, multiply by the radius of the circle:
Rotational Motion (2 of 2)
Torque
When the doorknob is pushed or pulled, the force rotates the
door about its hinge. The turning action depends on the product
of the force component perpendicular to the lever arm and the

8. lever arm. This quantity is called torque, τ, and its units are N
m.
Torque = Force perpendicular to lever arm x length of lever arm
τ=Fperpr
Newton's Law for Rotation
Newton's second law, F = ma, is for translation. For rotation the
Newton's law is τ= Iα.
In this equation
τ is the torque,
I is the rotational inertia of the rigid body, and
α its angular acceleration.
A pair of dumbbells that has its two masses farther away from
each other, is much more difficult to twist than one in which the
masses are close together. Rotational inertia, I, depends not
only on the mass of the object, but also on how this mass is
distributed about the axis of rotation. The farther the mass is
from the axis of rotation, the greater the rotational inertia.
Rotational Equilibrium
An object is in rotational equilibrium when the net torque acting

9. on the object is zero. This in turn means that the angular
acceleration is zero. Rotational equilibrium defines when an
object is in balance, and can also be used to define when a lever
will be able to raise a mass.
Week 3 Summary
This week covered forces in two dimensions, circular motion,
inclined planes, projectiles, and rotation. You learned how to
apply Newton's laws in situations with different types of
motion. Relate the concepts covered in this week to more real
life situations.
7 ROTATIONAL MOTION AND THE LAW OF GRAVITY
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7 ROTATIONAL MOTION AND THE LAW OF GRAVITY
Astronauts fall around the Earth at thousands of meters per
second, held by the centripetal force provided by gravity.
Rotational motion is an important part of everyday life. The

10. rotation of the Earth creates the cycle of day and night, the
rotation of wheels enables easy vehicular motion, and modern
technology depends on circular motion in a variety of contexts,
from the tiny gears in a Swiss watch to the operation of lathes
and other machinery. The concepts of angular speed, angular
acceleration, and centripetal acceleration are central to
understanding the motions of a diverse range of phenomena,
from a car moving around a circular racetrack to clusters of
galaxies orbiting a common center.
Rotational motion, when combined with Newton's law of
universal gravitation and his laws of motion, can also explain
certain facts about space travel and satellite motion, such as
where to place a satellite so it will remain fixed in position over
the same spot on the Earth. The generalization of gravitational
potential energy and energy conservation offers an easy route to
such results as planetary escape speed. Finally, we present
Kepler's three laws of planetary motion, which formed the
foundation of Newton's approach to gravity.
7.1 ANGULAR SPEED AND ANGULAR ACCELERATION
In the study of linear motion, the important concepts are
displacement Ax, velocity v, and acceleration a. Each of these
concepts has its analog in rotational motion: angular
displacement Δθ, angular velocity ω, and angular acceleration a.

11. The radian, a unit of angular measure, is essential to the
understanding of these concepts. Recall that the distance s
around a circle is given by s = 2πr, where r is the radius of the
circle. Dividing both sides by r results in s/r = 2π. This quantity
is dimensionless because both s and r have dimensions of
length, but the value 2π corresponds to a displacement around a
circle. A half circle would give an answer of p, a quarter circle
an answer of π/2. The numbers 2π, π, and π/2 correspond to
angles of 360°, 180°, and 90°, respectively, so a new unit of
angular measure, the radian, can be introduced, with 180° = π
rad relating degrees to radians.
The angle θ subtended by an arc length s along a circle of radius
r, measured in radians counterclockwise from the positive x-
axis, is
The angle θ in Equation 7.1 is actually an angular displacement
from the positive x-axis, and s the corresponding displacement
along the circular arc, again measured from the positive x-axis.
Figure 7.1 illustrates the size of 1 radian, which is
approximately 57°. Converting from degrees to radians requires
multiplying by the ratio (π rad/180°). For example, 45° (π
rad/180°) = (π/4) rad.

12. Tip 7.1 Remember the Radian
Equation 7.1 uses angles measured in radians. Angles expressed
in terms of degrees must first be converted to radians. Also, be
sure to check whether your calculator is in degree or radian
mode when solving problems involving rotation.
For very short time intervals, the average angular speed
approaches the instantaneous angular speed, just as in the linear
case.
The instantaneous angular speedω of a rotating rigid object is
the limit of the average speed Δθ/Δt as the time interval Δt
approaches zero:
SI unit: radian per second (rad/s)
We take ω to be positive when θ is increasing (counterclockwise
motion) and negative when θ is decreasing (clockwise motion).
When the angular speed is constant, the instantaneous angular
speed is equal to the average angular speed.
EXAMPLE 7.1 Whirlybirds
Goal Convert an angular speed in revolutions per minute to

13. radians per second.
Problem The rotor on a helicopter turns at an angular speed of
3.20 × 102 revolutions per minute. (In this book, we sometimes
use the abbreviation rpm, but in most cases we use rev/min.) (a)
Express this angular speed in radians per second. (b) If the rotor
has a radius of 2.00 m, what arclength does the tip of the blade
trace out in 3.00 × 102 s?
Strategy During one revolution, the rotor turns through an angle
of 2π radians. Use this relationship as a conversion factor.