Name: Department: Student ID #: Notice ˆ + ( 1) points per correct (incorrect) answer ˆ No penalty for an unanswered question ˆ Fill the lank ( ) with ( ) if the statement is correct (incorrect) ˆ : corrections to an incorrect answer ˆ : solution ˆ Textook: Walker, Halliday, Resnick, Principles of Physics, Tenth Edition, John Wiley & Sons (14) 7-1 Kinetic Energy 1 ( ) 4 The weight of an oject on the moon is one-sixth of its weight on the Earth (a) ( ) The ratio of the magnitude of the gravitational force on a ody on the Earth to that of the same ody on the moon is 6:1 () ( ) The ratio of the kinetic energy of a ody on the Earth moving with speed V to that of the same ody moving with speed V on the moon is 1:1 5 The kinetic energy of an oject of mass m moving with the velocity V is K 1 mv (a) ( ) The dimension for kinetic energy is [K] [M][L] [T ] () ( ) The dimension for work is [W ] [M][L] [T ] The work W done y a constant force F applied to an oject during the displacement d is W F d F d cos θ, where θ is the angle etween F and d (a) ( ) If the magnitudes of F and d are fixed, then W has the maximum value F d at θ () ( ) If the magnitudes of F and d are fixed, then W has the minimum value F d at θ π (c) ( ) If the magnitudes of F and d are fixed, then W if F and d are perpendicular 3 ( ) Work has the SI unit of the joule, the same as kinetic energy N m is an equivalent unit 1 J 1 kg m /s 1 N m (c) ( ) If there is a difference in the kinetic energy and it is positive, K K K 1 >, then the net external force has done work to the oject etween during the time interval [t 1, t ] (d) ( ) Kinetic energy is a scalar (e) ( ) Work is also a scalar 6 ( ) A oy holds a weight W at arm s length for t His arm is aove the ground y a perpendicular distance L The work done y the force of the oy on the weight while he is holding it is vanishing ecause the displacement is zero 7- Work and Kinetic Energy 1 ( ) Department of Physics, Korea University Page 1 of 6
Camping equipment weighing mg is pulled across a frozen lake y means of a horizontal rope The coefficient of kinetic friction is µk The work W done y the campers in pulling the equipment y a straight distance D at constant velocity is W µk mgd An oject moves in a circle at constant speed (a) ( ) The work done y the centripetal force is zero ecause the centripetal force is perpendicular to the velocity () ( ) The kinetic energy is invariant ecause there is no work done on the oject 3 ( ) (a) ( ) The ratio of the speed is r m v1 v m1 () ( ) The stopping distance of each oject is Ki F where Ki is the kinetic energy of each oject Di (c) ( ) The stopping distance of each oject can also e expressed as Di vi ai where ai is a magnitude of the acceleration of each oject Thus the stopping distance is independent of the mass (d) ( ) If m1 4m, the ratio of the stopping distance of m1 to that of m is 1:1 5 A particle starts from rest at time t and moves along the x axis The net force on it is proportional to t A man pulls a sled along a rough horizontal surface y applying a constant force F at an angle θ aove the horizontal He has pulled the sled y a horizontal distance d The work done y the man is F d cos θ 4 Two ojects with masses, m1 and m, have the same kinetic energy and are oth moving to the right The same constant force F is applied to the left to oth masses Department of Physics, Korea University (a) ( ) The force is F (t) αmt, where α is a constant of dimension [α] [L][T ] 3 Page of 6
() ( ) The acceleration is () ( ) For a given d, this work ecomes the maximum when d is parallel to g a(t) αt Wg mgd (c) ( ) The velocity is 1 1 v(t) v + αt αt, (c) ( ) If d is opposite to g, then the work is negative and Wg mgd where v is the initial velocity (d) ( ) The work done y this time-dependent force is proportional to t4 : Z W F dx Z F vdτ Z t 1 ατ dτ (αmτ ) Z t 1 α m τ 3 dτ 1 4 α mt 8 (d) ( ) If d is perpendicular to g, then the work is vanishing A man pushes a crate of mass m y a distance of D upward along a frictionless slope that makes an angle of θ with the horizontal The speed of the crate decreases at a rate of a(< ) 6 ( ) The amount of work required to stop a moving oject is equal to the kinetic energy of the oject 7-3 Work Done y the Gravitational Force 1 Condiser a particle of mass m under the gravitational force (a) ( ) The net force on the crate is Fnet ma i (F cos θ mg sin θ) i, where i is the unit vector up the slope () ( ) The force exerted y the man is F m(g sin θ a) cos θ (c) ( ) The constant a is determined as a g sin θ F cos θ m (d) ( ) The work done y the man is Wman F D cos θ md(g sin θ a) (a) ( ) The work Wg done y the gravitational force Fg during the displacement d is Wg mgd cos φ, where φ (g, d) Department of Physics, Korea University (e) ( ) The work done y the gravitational force is Wgravity mgd sin θ 7-4 Work Done y a Spring Force Page 3 of 6
1 A spring force is a variale force governed y Hooke s law: F spring kd, where k is the spring constant and d is the displacement from the relaxed state of the spring (a) ( ) The dimension of the spring constant is [k] [M][T ] () ( ) The unit of the spring constant is N/m kg/s (c) ( ) The work done y the spring force from position x i x i î to x f x f î is W spring xf x i xf x i F spring dx ( kx) dx ( kx)dx 1 k(x f x i ) Note that x is the displacement from the relaxed state (equilirium position) (d) ( ) The work done y the spring force from the relaxed position x i to x x î is W spring 1 kx Consider the case that a lock is displaced along the x axis from x i to x f while continuing to apply a force F applied in addition to the spring force to it (a) ( ) The change in the kinetic energy K is K K f K i W applied + W spring, where W applied and W spring are the work done y the applied force and that y the spring force, respectively () ( ) If v i and v f are oth zero, then W applied +W spring W applied W spring (c) ( ) When a certain ruer and is stretched a distance x, it exerts a restoring force of magnitude F Ax, where A is a constant The work done y a person in stretching this ruer and with a constant velocity from x to x L is 1 AL 7-5 Work Done y a General Variale Force 1 ( ) The work done y a variale force F (x) is W xf x i F (x)dx ( ) When a certain ruer and is stretched a distance x, it exerts a restoring force F ax + x, where a and are constants The work done in stretching this ruer and from x to x L is: 3 Consider a force W 1 al + 1 3 L3 F f v, where f and are constants We want to find v(t) when v(t ) v (a) ( ) The physical dimensions of f and are [f ] [M][L][T ], [] [M][T ] 1 () ( ) The equation of motion can e written as m dv dt v + f Department of Physics, Korea University Page 4 of 6
(c) ( ) By integration, we find that m which leads to t v dv dτ v V f, m t log v f v f (d) ( ) Hence, we find that v(t) f + ( v f ) e m t (e) ( ) As t, the exponential factor vanishes: e m t Therefore, the terminal velocity ecomes v terminal lim t v(t) f 4 A particle moving along the x axis is acted upon y a single force F F e kx, where F and k are constants The particle is released from rest at x It will attain a maximum kinetic energy (a) ( ) The force is positive definite Therefore, there is no limit in x () ( ) The work done y the force from to can e computed as W dxf (x) F e kx dx F k (c) ( ) According to the work-kinetic energy theorem, the work done y the net force is the increment of the kinetic energy Because the particle was initially at rest, we find that K F k 7-6 Power 1 Power is defined y the work done per unit time (a) ( ) The SI unit of power is the joule per second This unit is used so often that it has a special name, the watt (W): 1 watt 1 W 1 J/s () ( ) The instantaneous power is defined y the time derivative of work: P dw dt (c) ( ) Because the differential of work can e expressed as dw F dx, power can also e computed as P F dx dt F v A particle starts from rest and is acted on y a net force that does work with the time-dependent power proportional to t The speed of the particle is proportional to t: (a) ( ) The power is P αt, where α is a constant of dimension [α] [M][L] [T ] 4 () ( ) The kinetic energy of the particle K is K P dt t ατdτ 1 αt (c) ( ) According to the definition of the kinetic energy, we find that Thus mv αt v α m t Department of Physics, Korea University Page 5 of 6
3 A constant force F is the only force on a crate of mass m that starts from rest Consider the instant the crate has gone y the distance D (a) ( ) The crate experiences a constant acceleration: Prolems a F m ( ) The speed at that instant is v F D ad m ( ) The power is P F v F F D m 1 A lock of mass m is dropped onto a relaxed vertical spring that has a spring constant of k The lock ecomes attached to the spring and compresses the spring y a distance d efore momentarily stopping Answer the following questions for the period that the spring is eing compressed Assume that friction is negligile (a) ( ) The gravitational force and the displacement have the same direction () ( ) The work done y the gravitational force is W g mgd (c) ( ) The spring force and the displacement is opposite (d) ( ) The work done y the spring force is W spring 1 kd (e) ( ) The kinetic energy of the lock just efore it hits the spring is K 1 kd mgd (f) ( ) d depends on the speed v of the lock just efore the hit as d mg k [ ] 1 + 1 + kv mg (g) ( ) If we release the lock with vanishing initial velocity on the plate, then d is d mg k (h) If the speed at impact is douled, then the maximum compression d of the spring is d mg k [ 1 + 1 + 4kv mg ] Department of Physics, Korea University Page 6 of 6