Resource Lesson
Forces Acting at an Angle
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The equations used to calculate the horizontal and vertical components of a force
F
acting at an angle
θ
measured from the positive x-axis are:
If the angle given is actually a reference angle,
α
, to the nearest x-axis instead of the directional angle
θ
(which is always measured counterclockwise from the positive x-axis), you must decide whether
F
_{x}
and
F
_{y}
are negative. Remember that forces are measured in newtons.
In the example shown above, both the x-component and the y-component of
F
are pointing in negative directions. Usually, the magnitude of the component is labeled without the use of a negative sign (-) since it is conveyed graphically by the direction of the component's vector. The following diagram is provided to assist you in remembering which functions are positive in which quadrants.
You can also recall these relationships by remembering that when the
x-component of the reference angle points along the positive x-axis, cosine and its reciprocal function secant are positive
y-component of the reference angle points along the positive y-axis, sine and its reciprocal function cosecant are positive.
Let's work a few examples.
What are the magnitudes of the x- and y-components of the force vector, F = (10.0 nts, 75º)?
What is the magnitude of the x-component of the force vector F = (10.0 nts, 105º)?
What is the magnitude of the y-component of the force vector F = (10.0 nts, 255º)?
Problems involving "diagonal" forces
Let's begin by examinine a force
F
pulling up at an angle
θ
on one side of an object which is located on a rough horizontal plane. A freebody diagram of this situation is shown below.
Our first step will be to resolve, or break down, the diagonal force into its horizontal and vertical components. These components,
F
_{x}
and
F
_{y}
are shown in the next diagram.
Since the object is in
vertical equilibrium
, the relationship
Σ
F
_{y}
= 0 will produce the equation
N = mg - F sin θ
If
F
has been pushing down at angle
θ
on the object instead of pulling up at angle
θ
,
then the value for the normal would change to
N = mg + F sin θ.
If the object is also in
horizontal equilibrium
, that is, it is either at rest or moving with a constant velocity towards the right, the relationship
Σ
F
_{x}
= 0 yields the equation
f = F cos θ where f = µN.
If the object were
accelerating to the right
, the equation that could be used to calculate the object's horizontal acceleration would be
net F
_{x}
= ma
T cos θ - f = ma where f = µN.
Notice that the magnitude of the acceleration will be changed by the value of
f
since it is impacted by the value of the normal. Hence, it is easier to accelerate an object by pulling up on it (decreasing the normal force) than by pushing down on it (increasing the normal force).
Let's work some additional examples.
Refer to the following information for the next two questions.
Suppose a 3-kg block is being pushed against a wall by a force F = 15 N acting at an angle of 30º to the horizontal.
What is the normal force of the wall on the block?
What is the magnitude of the friction present between the block and the wall?
Refer to the following information for the next five questions.
Using the system shown below, what would be the maximum mass, M, that can be supported by the vertical rope if the coefficient of static friction between the table and the 10-kg block is 0.4?
We will start our solution by examining freebody diagrams and the static equilibrium equations:
Σ
F
_{x}
= 0 and
Σ
F
_{y}
= 0. Since tensions are 3rd law forces we can find the tension on either side of a "rope" and it will be constant throughout the rope.
T
_{3}
= Mg
T
_{2}
= T
_{1}
cos 60
T
_{1}
sin 60 = T
_{3}
f
_{s}
= µN
N = mg
f = T
_{2}
What is the maximum, or critical, value for the static friction present between the 10-kg block and the table?
What is the value of T
_{2}
?
What is the value of T
_{1}
?
What is the value of T
_{3}
?
What is the value of the suspended mass, M?
Other common applications of forces applied at angles are objects moving along incline planes, simple pendulums, conical pendulums, and systems of bodies with knots and pulleys. Each of these topics are included in their own lessons which are hyperlinked in the resource lessons listed under
Related Documents
at the bottom of the page.
Note that when working with "frictionless, massless" pulleys, the tensions on either side of the pulley are the same and should be labeled identically.
NO COMPONENTS
should be calculated for ropes as they go into or come out of frictionless, massless pulleys regardless of any angles present.
Related Documents
Lab:
Labs -
Coefficient of Friction
Labs -
Coefficient of Friction
Labs -
Coefficient of Kinetic Friction (pulley, incline, block)
Labs -
Conservation of Momentum in Two-Dimensions
Labs -
Falling Coffee Filters
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Force Table - Force Vectors in Equilibrium
Labs -
Inelastic Collision - Velocity of a Softball
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Inertial Mass
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LabPro: Newton's 2nd Law
Labs -
Loop-the-Loop
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Mass of a Rolling Cart
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Moment of Inertia of a Bicycle Wheel
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Relationship Between Tension in a String and Wave Speed
Labs -
Relationship Between Tension in a String and Wave Speed Along the String
Labs -
Static Equilibrium Lab
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Static Springs: Hooke's Law
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Static Springs: Hooke's Law
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Static Springs: LabPro Data for Hooke's Law
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Terminal Velocity
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Video LAB: A Gravitron
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Video LAB: Ball Re-Bounding From a Wall
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Video Lab: Falling Coffee Filters
Resource Lesson:
RL -
Advanced Gravitational Forces
RL -
Air Resistance
RL -
Air Resistance: Terminal Velocity
RL -
Freebody Diagrams
RL -
Gravitational Energy Wells
RL -
Inclined Planes
RL -
Inertial vs Gravitational Mass
RL -
Newton's Laws of Motion
RL -
Non-constant Resistance Forces
RL -
Properties of Friction
RL -
Springs and Blocks
RL -
Springs: Hooke's Law
RL -
Static Equilibrium
RL -
Systems of Bodies
RL -
Tension Cases: Four Special Situations
RL -
The Law of Universal Gravitation
RL -
Universal Gravitation and Satellites
RL -
Universal Gravitation and Weight
RL -
What is Mass?
RL -
Work and Energy
Worksheet:
APP -
Big Fist
APP -
Family Reunion
APP -
The Antelope
APP -
The Box Seat
APP -
The Jogger
CP -
Action-Reaction #1
CP -
Action-Reaction #2
CP -
Equilibrium on an Inclined Plane
CP -
Falling and Air Resistance
CP -
Force and Acceleration
CP -
Force and Weight
CP -
Force Vectors and the Parallelogram Rule
CP -
Freebody Diagrams
CP -
Gravitational Interactions
CP -
Incline Places: Force Vector Resultants
CP -
Incline Planes - Force Vector Components
CP -
Inertia
CP -
Mobiles: Rotational Equilibrium
CP -
Net Force
CP -
Newton's Law of Motion: Friction
CP -
Static Equilibrium
CP -
Tensions and Equilibrium
NT -
Acceleration
NT -
Air Resistance #1
NT -
An Apple on a Table
NT -
Apex #1
NT -
Apex #2
NT -
Falling Rock
NT -
Falling Spheres
NT -
Friction
NT -
Frictionless Pulley
NT -
Gravitation #1
NT -
Head-on Collisions #1
NT -
Head-on Collisions #2
NT -
Ice Boat
NT -
Rotating Disk
NT -
Sailboats #1
NT -
Sailboats #2
NT -
Scale Reading
NT -
Settling
NT -
Skidding Distances
NT -
Spiral Tube
NT -
Tensile Strength
NT -
Terminal Velocity
NT -
Tug of War #1
NT -
Tug of War #2
NT -
Two-block Systems
WS -
Advanced Properties of Freely Falling Bodies #1
WS -
Advanced Properties of Freely Falling Bodies #2
WS -
Calculating Force Components
WS -
Charged Projectiles in Uniform Electric Fields
WS -
Combining Kinematics and Dynamics
WS -
Distinguishing 2nd and 3rd Law Forces
WS -
Force vs Displacement Graphs
WS -
Freebody Diagrams #1
WS -
Freebody Diagrams #2
WS -
Freebody Diagrams #3
WS -
Freebody Diagrams #4
WS -
Introduction to Springs
WS -
Kinematics Along With Work/Energy
WS -
Lab Discussion: Gravitational Field Strength and the Acceleration Due to Gravity
WS -
Lab Discussion: Inertial and Gravitational Mass
WS -
net F = ma
WS -
Practice: Vertical Circular Motion
WS -
Ropes and Pulleys in Static Equilibrium
WS -
Standard Model: Particles and Forces
WS -
Static Springs: The Basics
WS -
Vocabulary for Newton's Laws
WS -
Work and Energy Practice: Forces at Angles
TB -
Systems of Bodies (including pulleys)
TB -
Work, Power, Kinetic Energy
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