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Work & Energy Methods
Ken Youssefi MAE 1
So far we have solved problems using Newtons 2nd
law, force is related to acceleration. Acceleration is then integrated to obtain velocity and position
Using Newtons 2nd law together with the principles of kinematics allows us to obtain two other methods of analysis; method of work and energy and method of impulse and momentum.
For these methods to apply, the force has to be a function of position (work and energy method) or time (impulse and momentum)
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Work & Energy Methods
Ken Youssefi MAE 2
The methods provide no information about the acceleration.
The methods deal directly with velocity rather than acceleration.
The methods deal with scalar quantity rather than vectors. So the problem formulation and solution are simpler.
Forces that do no work are ignored.
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Definition of Work
Ken Youssefi MAE 3
Work = force x displacement P
d
Units:
1 ft - lb = (1 ft) (1 lb) = (.3048 m) (4.448 N) = 1.356 Joule (J)
ft lb English (US customary units)
N m = (Joule) Metric (SI units)
Work = (P cos) x d
The vertical component = (P sin) of force P does no work. This force is ignored
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Definition of Work
Ken Youssefi MAE 4
The work U done on an object as its center of mass moves from a position r1 to a position r2is defined in terms of the external force F on the object and the displacement dr of its center of mass.
2
112
r
rrF dU
Whatever the path, only the force component that is aligned with the path contributes to the work done
2
1
)cos(12s
sdsFU
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Work Done by Gravity
Ken Youssefi MAE 5
When the force is defined by its rectangular coordinates , the expression for the work done is:
Fx = 0, Fy = -W = -mg, and Fz = 0
The work done by gravity is positive when y < 0, that is, when the body moves down
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Work Done by a Linear Spring
Ken Youssefi MAE 6
Body A is attached to a fixed frame B by a spring, the spring is unstretched at Ao
k is the spring stiffness in N/m or lb/ft
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Power and efficiency
Ken Youssefi MAE 7
Power is defined as the time rate at which work is done.
Substitute the scalar product F dr for dU: .
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Power and efficiency
Ken Youssefi MAE 8
Units of Power
SI system - watt English system - horsepower
Mechanical efficiency
The energy loss due to friction (heat) causes the power to be less than input power. So the efficiency is always less than 1.
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Kinetic Energy of a ParticlePrinciple of Work and Energy
Ken Youssefi MAE 9
Consider a particle m acted on by a force F, moving along a path.
Tangential component does work
At A1, s = s1 and v = v1 and at A2, s = s2 and v = v2
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Kinetic Energy of a ParticlePrinciple of Work and Energy
Ken Youssefi MAE 10
The left side of the equation is the work done moving from position 1 to 2, U1-2
The right side of the equation is the change in kinetic energy of the particle
Units of kinetic energy
Principle of Work and Energy
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Example
Ken Youssefi MAE 11
A car weighing 4000 lb is moving down a 5o incline at a speed of 60 mi/h when the brakes are applied, causing a constant braking force of 1500 lb (applied by the road on the tires). Determine the distance traveled by the car as it comes to stop.
Kinetic energy
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Example
Ken Youssefi MAE 12
Principle of work and Kinetic energy
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Example
Ken Youssefi MAE 13
Two blocks are joined by a non-stretching cable. If the system is released from rest, determine the velocity of the block A after it has moved 2 m. Assume the coefficient of kinetic friction is 0.25 and the pulley is weightless and frictionless
Principle of work and kinetic energy for block A
Fc = cable force, FA = friction force
FBD of block A
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Example
Ken Youssefi MAE 14
Principle of work and kinetic energy for block B
Add equations (1) and (2) to eliminate Fc
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Example
Ken Youssefi MAE 15
A spring is used to stop a 60 kg package sliding on a horizontal surface. The spring has a constant stiffness of 20 kN/m and is held by a cable so that it is initially compresses 120 mm. Knowing that the package has a velocity of 2.5 m/s and that the maximum additional deflection of the spring is 40 mm, determine a) the coefficient of kinetic friction, b) the velocity of the package as it passes again through the position shown.
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Example
Ken Youssefi MAE 16
Motion from position 1 to position 2
Work done by the friction force
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Example
Ken Youssefi MAE 17
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Example
Ken Youssefi MAE 18
Motion from position 2 to position 3
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Example
Ken Youssefi MAE 19
A 2000 lb car starts at position 1 and moves down along the track (neglect friction). Determine a) the force exerted by the track on the car at position 2, where the radius of curvature of the track is 20 ft, b) determine the minimum safe value of the radius of curvature at position 3
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Example
Ken Youssefi MAE 20
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Example
Ken Youssefi MAE 21
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Potential Energy Gravity Force
Ken Youssefi MAE 22
Consider the work down by the weight W moving along a curved path from point A1 to A2.
The work done by W is independent of the path.
Wy is called the potential energy of the body with respect to gravity force W, it is denoted by Vg
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Potential Energy Linear Spring
Ken Youssefi MAE 23
Consider a body attached to a spring and moving from position A1 to A2.
k x2 is called the potential energy of the body with respect to elastic
force F, it is denoted by Ve
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Conservative Forces
Ken Youssefi MAE 24
If the work done by a force (in moving a particle) is independent of path, that force is called a conservative force.
Work done by force F as the particle moves from A1 to A2.
The function V is called the potential energy or potential function of F
If point A2 coincides A1 , that is if the particle describes a closed path the work done is zero.
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Conservative Forces
Ken Youssefi MAE 25
Consider two point A (x, y, z) and A (x + dx, y + dy, z + dz) on the path. The work done dU can be written:
Substituting for dU in terms of rectangular components and using the definition of the differential of a function of several variable, we have
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Non-Conservative Force
Ken Youssefi MAE 26
Sliding friction is a very common example of a non-conservative force, because the amount work done depends on the path taken.
Sliding friction
mgLdsmgU kL
k 012
L is the length of the curve
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Conservation of Energy
Ken Youssefi MAE 27
When a body moves under the action of conservative forces, the principle of work and energy can be modified as follows
Principle of work and energy
Kinetic energy
Work done by conservative forces Potential energy
When a particle moves under the action of conservative forces, the sum of the kinetic and potential energy remains constant
Conservation of Energy equation
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Conservation of Energy
Ken Youssefi MAE 28
Mechanical energy = T + V
The kinetic energy has the same value at any two points with the same elevation.
Assume friction is negligible (friction force is a nonconservative force)
The body has the same velocity at A, A, and A.
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Example
Ken Youssefi MAE 29
A 20 lb collar slides along a vertical rod (neglect friction). The spring attached to the collar has an undeformed length of 4 in. and a stiffness of 3 lb/in. If the collar is released from rest in position 1, determine its velocity after it has moved 6 inch to position 2.
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Example
Ken Youssefi MAE 30
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Example
Ken Youssefi MAE 31
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Example
Ken Youssefi MAE 32
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Comments on Work & Energy and Conservation of energy Methods
Ken Youssefi MAE 33
The methods provide no information about the acceleration.
The methods deal directly with velocity rather than acceleration.
The methods deal with scalar quantity rather than vectors. So the problem formulation and solution are simpler.
Forces that do no work are ignored.
Conservation of energy method is useful only if conservative forces are applied (gravity and spring forces)
