Review: Newton s Laws, Work, Energy, and Power These are selected problems that you are to solve independently or in a team of 2-3 in order to better prepare for your Exam#2 1
Problem 1: Inclined Plane Two blocks are attached via a rope (no mass), guided over a pulley (no mass). Mass m 1 can slide on an inclined plane with a coefficient of kinetic friction of 0.10 and a coefficient of static friction of 0.40. a) If m 1 =5.0kg initially slides to the right, what is the acceleration of the system? b) What is the tension in the rope? c) If the system starts from rest, for what range of masses m 1 will the system not accelerate? d) Now assume that there is no friction. How large does m 1 have to be such that the system does not accelerate? Problem 2: Bear Sling You are camping at the rim of the Grand Canyon, where bears are abundant. To protect your food (and yourself) from them, you decide to abide by regulations and hang your food in a backpack from a bear sling, as shown in the picture. The backpack has a mass m, the distance between the two trees is l and the height above the ground of the attachments of the rope to the trees is h. Consider the rope massless and the pulleys ideal. a) Initially, as the backpack is on the ground, what is the force F you need to pull with (see figure) to lift the backpack of the ground? b) Find a formula for the force F which describes the force for a situation, when the backpack is a distance d above the ground. Note: This formula will depend on m, g, h, d and l. c) For m=10.0kg, h=5.0m, d=4.0m (i.e. the rope sags by h-d=1.0m) and l=8.0m, what is F. d) Use your formula from b (or a physical argument) to argue if you can pull the rope with sufficient force to have no sagging at all, or if this is impossible. e) How do your answers change if the backpack is not in the middle of the rope but more to the right? 2
Problem 3: Frictionless table A flat puck (mass M) is rotated in a circle on a frictionless table top, and is held in this orbit by a light cord (no mass) which is connected to a dangling cube (mass m) through the central hole as shown. a) If the radius of rotation has the constant value R, draw a Free Body Diagram for the puck. b) For a state of constant R, draw a Free Body Diagram for the dangling cube. c) Derive an equation giving the speed of the puck (v) and including only the constant values of m, M, R and g. d) If M=m=1kg and v=0.35m/s, calculate R in meters. Problem 4: Work and Energy A box of mass m=1.0kg is at rest at point A. AB is a frictionless plane inclined at 30 with respect to the horizontal. BCD is a horizontal plane. At point B there is a smooth transition between the motion on the two planes (i.e. m s speed just before arriving at B is equal to the horizontal speed just after leaving B). At point C the mass makes contact with an uncompressed spring of spring constant k=480n/m. C is 4m from B. The coefficient of kinetic friction on the segment from B to C is 0.15. a) What is the acceleration (in m/s 2 ) of the box at point A? b) If the box takes 3s to travel from A to B, what is its speed (in m/s) at point B? c) What is the work (in Joules) that gravity does on the box during the motion from A to B? d) What is the kinetic energy (in Joule) of the box at B? e) What is the work (in Joule) that friction does on the box during the motion from B to C? f) What is the kinetic energy (in Joule) of the box at C? g) In order to bring the box to a full rest, by how much does the length of the spring get compressed (in meters)? 3
Problem 5: Pulley A piano of mass M is lifted using the shown pulley mechanism. Assume that all ropes shown in the picture are massless and exactly vertical. a) What is the tension (force) F T4? b) What is the tension (force) F T1? c) What is the tension (force) F T2? d) What is the tension (force) F? e) What is the tension (force) F T3? Problem 6: Antarctica Two snow cats tow a housing unit to a new location at McMurdo Base, Antarctica. The sum of the forces F A and F B exerted on the unit by the horizontal cables is parallel to the line L. F A =6500N. Vector quantities are shown in bold letters. a) How large are the force components of F A, which pull (1) parallel to L and (2) perpendicular to L? b) How large are the force components of F B, which pull (1) parallel to L and (2) perpendicular to L? c) How large is F B? d) How large is the total force F A + F B? e) If the housing unit is being pulled at a constant velocity of 25m/s on a horizontal snow field, what is the magnitude of the friction force between the snow on which the housing unit slides and the housing unit? Explain. 4
Problem 7: Ball on Cord (1) A ball is attached to a massless horizontal cord of length L whose other end is fixed. Initially it is at rest (1). Consider the ball to be a point object of mass m, which is attracted downwards by the Earth s gravitational acceleration g. (3) Note: is the plane of rotation horizontal or vertical? a) Explain in one phrase (max. 3 words) which physical concept allows you to calculate the speed at point (2) of its path, considering the quantities given above. (2) b) If the ball is released, what will be its speed v 2 at point (2) of its path? (Derive the formula for v 2 ) c) A peg is located a distance h directly below the point of attachment of the cord. If h=0.80l, what will be the speed of the ball when it reaches point (3)? Problem 8: Ball and Spring A ball of mass m=2.60kg, starting from rest at y 1, falls a vertical distance h=55.0cm before striking a vertical uncompressed coiled spring at y 2, which it then compresses an amount y 3 =-Y=-15.0cm. (Note: the zero of y is chosen at y 2 to make it easier!) a) What is the potential energy U 1 of the ball at y=y 1? b) What is the kinetic energy K 1 of the ball at y=y 1? c) What is the velocity of the ball at y=y 2? d) What is the potential energy U 2 of the ball at y=y 2? e) What is the kinetic energy K 2 of the ball at y=y 2? f) Between y=y 2 and y=y 3, the kinetic energy K 2 and gravitational potential energy U G are converted into elastic potential energy U E. How much elastic potential energy is stored in the spring at y=y 3? g) Using the result from f), calculate the spring constant k of the spring. 5
Problem 9: Blocks and Forces A block of mass m 1 is at rest on a smooth horizontal surface, connected by an un-stretchable cord that passes over a pulley to a second block of mass m 2, which hangs vertically. Ignore friction and the masses of the pulley and cord. (a) Draw a free-body diagram for the block of mass m 1, showing all forces exerted on the block? (b) Draw a free-body diagram for the block of mass m 2, showing all forces exerted on the block? (c) Derive the formula for the acceleration of the system. (d) Derive the formula for the tension in the cord. Problem 10: Parachute A person of mass m=65kg just jumped out of an airplane and is accelerating vertically towards the ground, while the parachute remains closed. The (constant) terminal velocity of the person, reached after the acceleration is complete, is v T =360km/h. Assume the parachute to be massless. (a) After reaching v T, the person descents for 30 seconds prior to opening the parachute. What work does air resistance do on the person during this time? (b) What is the power that air resistance dissipates during this time? (c) Then, the person deploys the parachute and subsequently slows down to a speed of 18km/h over a vertical length of 500m. Calculate the work done by air resistance on the person during this 500m deceleration. (d) When the person touches down on a sandy beach shortly afterwards, the legs must do work in order to affect the slow down to zero speed. Assuming a constant force, if the person sinks 30cm into the sand, calculate the force required for the touchdown. 6