Work and Kinetic Energy I

Transcription

1 Work and Kinetic Energy I Scalar Product The scalar product of any two vectors A and B is a scalar quantity equal to the product of the magnitudes of the two vectors and the cosine of the angle φ between them A B ABcosφ The unit vectors i, j and k lie in the positive x, y and z directions rectangle coordinate system. Therefore, i i = j j = k k = 1 i j = i k = j k = 0 If vector A and B is defined as A = A xi + A yj + A zk and B = B xi + B yj + B zk, then and A A = A x 2 + A y 2 + A z 2 A B = A x B x + A y B y + A z B z Ex7.2) A = 2i + 3j and B = -i + 2j. (a) Find the scalar product of A B. (b) Find the angle φ between A and B. Work done by Constant Force The work W done on an object by an agent exerting a constant force on the object is the product of the component of the force in the direction of the displacement and the magnitude of the displacement. W = F d = Fdcosφ - Unit of work is expressed in Joules(J) - If you apply 1N of force over a distance of 1meter, then you are doing 1Joule of work. FN Fapp φ Fapp cos φ Fg = mg d= distance traveled W = F app d = Fdcosφ - Work done by normal force F N is zero; W=F N dcos90º =0 - Work done by gravitational force F g (=weight) is zero; W=F g dcos90º = 0

2 1. A boy exerts a horizontal force of 10N on a box at a distance of 25m. How much work did he do on the box? a) 200J b) 250J c) 300J d) 350J 2. A girl exerts a force of 35N at an angle of 35 above the horizontal along the handle of her wagon as she pulls it 25m across a level sidewalk. How much work does she do on her wagon? a) 717J b) 156J c) 134J d) 120J 3. An object of mass 3kg is dropped from a height of 4m above the ground. Find the work done by gravity(w g) throughout the motion until it reaches the ground. a) 117.6J b) J c) 29.4J d) -29.4J 4. A box on the floor is lifted by man. If the mass of the box is 60kg, and he lifts it 2m from the ground at a constant speed, i) how much work did the man do on the box? FL ii) how much work did gravity do on the box? 5. A crate of mass 5kg is released from the top of frictionless ramp. The angle of the incline is 18. i) What is the force that does work on the crate? ii) Find the work done on the crate over distance of 4m along the ramp. a) 34.4J b) 47.3J c) 51.2J d) 60.6J

3 Work-Kinetic Energy Theorem If you apply a force over a certain distance, there will be a change in kinetic energy vi distance d vf Fapp Fapp Ki + F d cosφ = Kf 1 2 mvi 2 +F app dcosφ = 1 2 mvf 2 K i + ΣW = K f : Work-Kinetic Energy Theorem where K= 1 2 mv2 6. A 150g mass has a velocity v=(2i + 6j) m/s at a certain instant. What is its kinetic energy? ( Use vector dot product to find the speed. That is, v v=v 2 ) 7. How much force is needed to accelerate a 1300kg car from rest to a speed of 20m/s in a distance of 80m? 8. Revisit question 4 using Work-Kinetic Energy Theorem A box on the floor is lifted by man. If the mass of the box is 60kg, and he lifts it 2m from the ground at a constant speed, i) how much work did the man do on the box? FL ii) how much work did gravity do on the box?

4 W= F d K = 1 2 mv2 K i + ΣW = K f f =µf N 9. An object of mass 2kg is moving on a frictionless surface with a constant speed of 8m/s. How much force is needed to stop the object over a distance of 10m? a) 3.9N b) 6.4N c) 8.3N d) 11.8N 10. A 4kg box is sliding on a surface with an initial speed of 8m/s. The coefficient of kinetic friction between the box and the floor is µ k =0.4. What is the speed of the box after it moved a distanced of 4m? (*~frictional force f=µf N, where F N=mg in this case ~*) a) 19.6m/s b) 14.2m/s c) 10.5m/s d) 5.7m/s 11. A 8kg box is released from rest on a 30 inclined frictionless plane and accelerates down the incline. If the distance from the block to the end of the ramp is 4m, what is the speed of the block when it reaches the bottom of the ramp? a) 6.3m/s b) 3.6m/s c) 1.6m/s d) 0.9m/s

5 W= F d K = 1 2 mv2 K i + ΣW = K f f =µf N 12. A 8kg box is released from rest on a 30 inclined rough plane and accelerates down the incline. The coefficient of friction µ k=0.54. i) If the distance from the block to the end of the ramp is 4m, what is the speed of the block when it reaches the bottom of the ramp? a) 3.3m/s b) 2.4m/s c) 1.6m/s c) 0.4m/s ii) The block will eventually slide off the ramp and continue on the surface. If the coefficient of kinetic friction on the surface was µ k=0.12, how far will the block travel until it stops? a) 2.21m b) 1.63m c) 1.07m d) 0.64m

6 W= F d K = 1 2 mv2 K i + ΣW = K f f =µf N 13. A 12kg crate is pushed up on a frictionless incline plane from the bottom to top at constant speed. The height of the incline is h=5m and the distance of the plane is d=14m. The angle of the incline is 21. How much work was done by the pushing force F p? That is, find W p. a) 590J b) 1200J c) -590J d) -1200J h=5m d=14m Fp Hint: Fp φ - Use K i + ΣW = K f - Work done by gravity is W g = F gdcosφ, where φ= θ Work done by pushing force W p= F pcos0 d. θ Fg