# WORK DONE BY A CONSTANT FORCE

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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 = = Fdcosθ F r d r F r d r 1

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? 2

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 ) 3

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 W total = W 1 + W 2 + W 3 + = ΣW i Also the total work can be calculated by first performing a vector sum of all the forces acting on an object to obtain F r total and then using the basic definition of work W total = (F total cosθ)d = F total dcosθ 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 N r exerted by the road; (ii) air resistance force, F r ; (iii) gravity mg r. Find air the total work done on the car as it travels a distance d along the road. 4

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 = F total /m Suppose the apple falls with initial speed v i, and after distance d has a final speed v f Constant-acceleration kinematics: v f2 = v i2 + 2ad 2(F total /m)d = v f2 - v i 2 F total d = ½ mv f2 ½ mv i2 = W total W total > 0 v f > v i ; W total < 0 v f < v i ; W total = 0 v f = v i ½ mv 2 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 5

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 W total = K = ½ mv f2 ½ mv i 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. 6

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(x 2 x 1 ) If a force has a value F 1 from x = 0 to x = x 1, then a different value F 2 from x 1 to x 2, W = F 1 x 1 + F 2 (x 2 x 1 ) 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 x 1 to x 2 is equal to the corresponding area between the force curve and the x axis 7

8 WORK TO STRETCH OR COMPRESS A SPRING The force exerted by a spring is F x = -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 kx 2 8

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 J. 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? 9

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 kg 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? 10

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 µ k N (= µ k mg) After sliding the box a distance d, the work you do is W = µ k mgd On releasing the box, it simply stays put friction does no work after you let go 11

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 W total = W AB + W BC + W CD + W DA W AB = W CD = 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 W total = 0 + (-mgh) mgh = 0 12

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 = -f k d = -µ k mgd on each segment Thus total work done by kinetic friction is W total = -4µ k mgd Thus it can be said that a conservative force is a force that does zero total work on any closed path 13

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 W total = W 1 + W 2 W 2 = -W 1 W total = W 1 + W 3 W 3 = -W 1 W 3 = W 2 14

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 µ k = 0.63, how much work is done by friction along each path 15

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 16

17 POTENTIAL ENERGY, U Kinetic energy is given by K = ½ mv 2 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 W c the corresponding potential energy U is changed according to W c = U i U f = -(U f U i ) = - 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 17

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 W c = mgy Change in potential energy: - U = U i U f = W c = mgy Rearranging gives: U i = mgy + U f U i > U f When falling from a height to ground level, U f = 0, thus U i = 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 18

19 POTENTIAL ENERGY AND SPRINGS The work required to stretch a spring from its equilibrium position a distance x is W = ½ kx 2 From definition of potential energy: W c = ½ kx 2 = U i U f U f = 0 because it is the potential energy when the spring is at equilibrium (x = 0) Thus the potential energy of a spring: U = ½ kx 2 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? 19

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