work done by a constant force - city, university of londondanny/2_work.pdf · suppose a 4970kg...

2. Work, Energy and Conservation of Energy

1

WORK DONE BY A CONSTANT

FORCE

• The definition of work, W, when a constant force (F) is in the direction of displacement (d) is W = Fd

• SI unit is the Newton-meter (Nm) = Joule, J• If you exert a force of 82N on a box and move it in the

direction of the force through a distance of 3.00m, then the work you have done is 82×3 = 246J

• W is zero if the displacement d is zero• Work when the angle between a constant force and the

displacement is θ is W = (Fcosθ)d• Fcosθ is the component of force in the direction of

displacement• Work can also be expressed as the dot product

between vectors and as W = = FdcosθFr

dr

dFrr

⋅


2

FORCE AT AN ANGLE TO THE

DISPLACEMENT: EXAMPLE

• In a gravity escape system (GES) , an enclosed lifeboat on a large ship is deployed by letting it slide down a ramp and then continuing in free fall to the water below. Suppose a 4970kg lifeboat slides a distance of 5.0m on a ramp, dropping through a vertical height of 2.5m. How much work does gravity do on the boat?


3

NEGATIVE WORK AND TOTAL

WORK (1)

• Work depends on the angle between the force and the displacement. This dependence gives rise to three distinct possibilities

• Work is positive if the force has a component in the direction of motion (-90° < θ < 90°)

• Work is zero if the force has no component in the direction of motion (θ = ±90°)

• Work is negative if the force has a component opposite to the direction of motion (90° < θ < 270°)


4

NEGATIVE WORK AND TOTAL

WORK (2)

• When more than one force acts on an object, the total work is the sum of the work done by each force separately

• Wtotal = W1 + W2 + W3 + … = ΣWi

• Also the total work can be calculated by first performing a vector sum of all the forces acting on an object to obtain and then using the basic definition of work

• Wtotal = (Ftotalcosθ)d = Ftotaldcosθ

• Example: A car of mass m coasts down a hill inclined at an angle φ below the horizontal. The car is acted on by three forces: (i) the normal force exerted by the road; (ii) air resistance force, ; (iii) gravity . Find the total work done on the car as it travels a distance dalong the road.

totalFr

Nr

airFr

gr

m


5

KINETIC ENERGY AND THE WORK-

ENERGY THEOREM (1)

• As an apple drops from a tree, the total force acting on it (due to gravity and air resistance), gives a constant downward acceleration of magnitude a = Ftotal/m

• Suppose the apple falls with initial speed vi, and after distance d has a final speed vf

• Constant-acceleration kinematics: vf2 = vi

2 + 2ad• 2(Ftotal/m)d = vf

2 - vi2

• Ftotald = ½ mvf2 – ½ mvi

2 = Wtotal

• Wtotal > 0 → vf > vi; Wtotal < 0 → vf < vi; Wtotal = 0 → vf = vi

• ½ mv2 is called the kinetic energy, K (also joules, J)• Kinetic energy of an object is due to its motion, and is

never negative. It is independent of the direction of motion, or the direction of any forces


6

KINETIC ENERGY AND THE WORK-

ENERGY THEOREM (2)

• Work-Energy Theorem states that the total work done on an object is equal to the change in its kinetic energy

• Wtotal = ∆K = ½ mvf2 – ½ mvi

2

• Example: A 4.1kg box of books is lifted vertically from rest a distance of 1.6m with a constant upward applied force of 52.7N. Find the work done by the applied force; the work done by gravity; the final speed of the box.

• Example: A boy exerts a force of 11.0N at 29° above the horizontal on a 6.4kg sled. Find the work done by the boy and the final speed of the sled after it moves 2.0m, assuming the sled starts with an initial speed of 0.5m/s and slides horizontally without friction.


7

WORK DONE BY A VARIABLE

FORCE

• For a constant force, the work done in moving an object a distance d is simply the area under the Force v Displacement graph W= F(x2 – x1)

• If a force has a value F1from x = 0 to x = x1, then a different value F2from x1 to x2, W = F1x1 + F2(x2 – x1)

• If a force varies continuously with position, we can approximate it with a series on constant values that follow the shape of the curve

• The work done by a force in moving an object from x1 to x2 is equal to the corresponding area between the force curve and the x axis


8

WORK TO STRETCH OR COMPRESS

A SPRING

• The force exerted by a spring is Fx = -kx

• So the force that must be exerted to hold it at the position x is +kx

• Thus the applied force versus position for a spring is a straight line of slope k

• Thus the work done to stretch or compress a spring a distance x from equilibrium is W = ½ kx2


9

WORK TO STRETCH OR COMPRESS

A SPRING: EXAMPLE

• In the field of nanotechnology, an atomic force microscope is used to view atomic-level pictures of surfaces

• It works in the same way as a record player, with a cantilever which is a thin silicon bar 250µm in length, supported at one end like a diving board, and with a sharp hanging point at the other end. When the point is pulled across the surface of a material (like a phonograph needed in a record player) individual atoms on the surface move the point up and down, deflecting the cantilever. These movements are converted using a laser into images.

• The work required to deflect a typical AFM cantilever by 0.1nm is 1.2×10-20J. What is the force constant of the cantilever, treating it as an ideal spring? How much work is required to increase the deflection on the cantilever from 0.1nm to 0.2nm?


10

POWER

• Power is a measure of how quickly work is done

• P = W/t (Joules/sec or watt, W)

• 1 watt = 1W = 1 J/s• Since W = Fd, then P = Fd/t

• Since speed is distance divided by time: P = Fv

• Another common unit of power is the horsepower (hp)• 1 horsepower = 1hp = 746W

• Example: To pass a slow moving truck, a car whose mass is 1.3×103kg has to accelerate from 13.4m/s to 17.9m/s in 3.00s. What is the minimum power required by the car to overtake the truck?


11

CONSERVATIVE AND

NONCONSERVATIVE FORCES

• Forces are classified according to whether they are conservative or nonconservative

• When a conservative force acts, the work it does is stored in the form of energy that can be released at a later time

• Gravity is a conservative force: lifting a box with mass m, an upward distance h and the work done on the box is mgh

• Releasing the box and allowing it to drop back to the floor, the work done by gravity is also mgh: this process gives the box an equivalent amount of kinetic energy

• The work done by a nonconservative force cannot be recovered by kinetic energy, instead it is converted to other forms of energy (e.g. heat)

• Kinetic friction is a form of nonconservative force• To slide a box across a surface at constant speed, you

must exert a force of magnitude µkN (= µkmg)• After sliding the box a distance d, the work you do is W

= µkmgd• On releasing the box, it simply stays put – friction does

no work after you let go


12

MOVING AN OBJECT IN A CLOSED

PATH

• Moving a box of mass m along the closed path above, the total work done by gravity is the sum of the work done on each segment of the path

• Wtotal = WAB + WBC + WCD + WDA

• WAB = WCD = 0 (work by gravity is zero; force at right angles to displacement)

• From B to C, gravity does negative work (displacement and force are in opposite directions)

• Gravity does positive work from D to A

• Thus Wtotal = 0 + (-mgh) + 0 + mgh = 0


13

MOVING AN OBJECT IN A CLOSED

PATH WITH FRICTION

• If a boxed is pushed around a closed horizontal path, the total work done by friction does not vanish

• Friction does negative work W = -fkd = -µkmgd on each segment

• Thus total work done by kinetic friction is

• Wtotal = -4µkmgd

• Thus it can be said that a conservative force is a force that does zero total work on any closed path


14

EXAMPLE OF CONSERVATIVE

FORCES

• If frictional forces can be ignored, a roller coaster car will have the same speed at points A and D, since they are at the same height

• Hence after any complete circuit of the track the speed of the car returns to its initial value

• It follows that the change in kinetic energy is zero (∆K = 0) for a complete circuit, and thus the work done by gravity is also zero

• The work done by a conservative force is zero on any closed path

• Wtotal = W1 + W2

• W2 = -W1

• Wtotal = W1 + W3

• W3 = -W1

• W3 = W2


15

PATHS AND FORCES: EXAMPLE

• A 4.57kg box is moved with constant speed from A to B along two paths as shown. Calculate the work done by gravity on each of the paths.

• The same box is pushed across a floor from A to B along path 1 and 2. If the coefficient of kinetic friction between the box and the surface is µk = 0.63, how much work is done by friction along each path


16

POTENTIAL ENERGY AND THE

WORK DONE BY CONSERVATIVE

FORCES

• Work is done when lifting an object onto a shelf

• Once on the shelf, the object has zero kinetic energy, as it did when on the floor

• But the work done in lifting the object has not been lost

• If the object fell from the shelf, gravity does the same amount of work as was done in lifting the object up to the shelf

• The work can be recovered in the form of kinetic energy

• Thus when the ball is lifted to a new position, there is an increase in potential energy, U, which can be converted to kinetic energy when the ball falls

• Potential energy is a storage system for energy – work done in increasing the separation between the object and the ground is stored in potential energy, which is never lost, and is only released when the object falls

• Work done against friction is never stored, only dispersed by heat or sound

• Only conservative forces have the potential energy storage system


17

POTENTIAL ENERGY, U

• Kinetic energy is given by K = ½ mv2 regardless of force involved

• But each different conservative force has a different expression for its potential energy

• When a conservative force does an amount of work Wcthe corresponding potential energy U is changed according to

• Wc = Ui – Uf = -(Uf – Ui) = -∆U (joule, J)

• The work done by a conservative force is equal to the negative of the change in potential energy

• When an object falls, gravity does positive work on it and its potential energy decreases

• When an object is lifted, gravity does negative work, and its potential energy increases

• This definition determines only the difference in potential energy between two points, not the actual potential energy


18

POTENTIAL ENERGY AND GRAVITY

• Potential energy near the Earth’s surface has a different definition

• If a person of mass m drops a distance y from a diving board into a pool, gravity does the work

• Wc = mgy

• Change in potential energy: -∆U = Ui – Uf = Wc = mgy

• Rearranging gives: Ui = mgy + Uf

• Ui > Uf

• When falling from a height to ground level, Uf = 0, thus Ui = mgy

• Now have a definition for the potential energy near the Earth’s surface: U = mgy

• Example: An 82kg mountaineer climbs up a 4301m high mountain. What is the change in gravitational potential energy as he gains the last 100m of altitude? Let U = 0 at a) sea level; b) at the top of the peak


19

POTENTIAL ENERGY AND SPRINGS

• The work required to stretch a spring from its equilibrium position a distance x is W = ½ kx2

• From definition of potential energy: Wc = ½ kx2 = Ui – Uf

• Uf = 0 because it is the potential energy when the spring is at equilibrium (x = 0)

• Thus the potential energy of a spring: U = ½ kx2

• Example: When a force of 120N is applied to a certain spring, it causes a stretch of 2.25cm. What is the potential energy of this spring when it is compressed by 3.5cm?

work done by a constant force - city, university of londondanny/2_work.pdf · suppose a 4970kg...

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