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Newton’s Laws (cont…) • Blocks, ramps, pulleys and other problems Physics 1D03 - Lecture 8 1 Example Two blocks connected by a rope are being pulled by a horizontal force FA. Given that FA=60 N, m1=12kg and m2=18kg, and that μk=0.1, find the tension in the rope between them and the acceleration of the system. T m1 m2 FA Physics 1D03 - Lecture 8 2 Solution Physics 1D03 - Lecture 8 3 Quiz Two blocks of mass m1 and m2 (m1>m2) are being pushed by a horizontal force FA over a frictionless surface. Consider the contact force between m1 and m2. FA m1 F12 m2 F21 A) F12 > F21 B) F12 = -F21 C) F12 > F21 D) F12 = F21 Physics 1D03 - Lecture 8 4 Example Two blocks of mass m1 and m2 (m1>m2) are being pushed by a horizontal force FA over a frictionless surface. Determine: a) the magnitude of the acceleration of the system b) the magnitude of the contact force between the blocks FA m1 m2 Physics 1D03 - Lecture 8 5 Solution Physics 1D03 - Lecture 8 6 Example A block of mass m=5kg is being pulled by two forces. F1 = 6N which acts along the positive x-axis, and F2 = 15N which acts at an angle of 135o to the positive x-axis. What is the acceleration of the block ? Physics 1D03 - Lecture 8 7 Example A block of mass m=2kg is suspended by two ropes as shown below. What are the tensions in each rope ? 30o 60o m Physics 1D03 - Lecture 8 8 Elevator go up, elevator go down • A person of mass 70kg is standing on a scale in an elevator at rest. What is her weight ? • What is her weight when the elevator is accelerating up at 5m/s2 ??? • What is her weight when the elevator is accelerating down at 5m/s2 ??? Physics 1D03 - Lecture 8 9 10 min rest Physics 1D03 - Lecture 8 10 Pulleys • To solve pulley problems, we assume that: 1) the pulley is frictionless 2) the pulley is massless • Hence, the force of tension on both sides of the pulley is the same Physics 1D03 - Lecture 8 11 Example • Find the acceleration of a system of two masses m=5kg and M=10kg. The angle θ=25o. No friction! • Also, find the tension, T, in the string. M m q There are two ways of solving the problem – find T first, or last. Physics 1D03 - Lecture 8 12 Solution Physics 1D03 - Lecture 8 13 Quiz Two masses m and M are connected as shown. The angle is θ. If there is friction on the incline, can we write the equation of motion without knowing the masses or the angle? M m q A) Yes, we can B) No, we can’t C) What’s an equation of motion ??? Physics 1D03 - Lecture 8 14 Example: A block of mass m1 on a rough horizontal surface is pulled with a force FA at an angle θ to the horizontal. A ball of mass m2 is connected to the other side, hanging over a lightweight frictionless pulley. The coefficient of friction is given by μk. Assume m1 will move to the right. Determine the acceleration of the system. FA θ m1 m2 Physics 1D03 - Lecture 8 15 Solution Physics 1D03 - Lecture 8 16 Atwood’s Machine Calculate the acceleration of the blocks. Assume : - no friction - massless rope and pulley - rope doesn’t stretch Plan: • free-body diagram for each mass • relate tensions, accelerations • use Newton’s second Law m1 Physics 1D03 - Lecture 8 m2 17 Forces on m1 T1 Forces on m2 T2 a2 a1 m2g m1g • • Tensions are equal (“ideal” pulley, light rope) Accelerations are equal in magnitude (why?), opposite in direction Physics 1D03 - Lecture 8 18 T T a a m2g m1g m1g T m1a T m2g m2a . . . m1 m2 g Eliminate T to get a m1 m2 a is proportional to g, but can be small (and easy to measure) Physics 1D03 - Lecture 8 19 10 min rest Physics 1D03 - Lecture 8 20 Motion in 2D • Constant acceleration in 2-D • Free fall in 2-D Physics 1D03 - Lecture 8 21 The Position vector r points from the origin to the particle. y yj path (x,y) r xi x The components of r are the coordinates (x,y) of the particle: r x i y j For a moving particle, r (t ), x(t), y(t) are functions of time. Physics 1D03 - Lecture 8 22 Components: Each vector relation implies 2 separate relations for the 2 Cartesian components. r xi y j (i, j are unit vectors) We get velocity components by differentiation: dr v dt dx dy i j dt dt vx i v y j the unit vectors are constants Physics 1D03 - Lecture 8 23 Constant Acceleration + Projectile Motion If a is constant (magnitude and direction), then: v (t ) vo a t 2 1 r (t ) vo t 2 a t Where vo is the initial value at t = 0. In 2-D, each vector equation is equivalent to a pair of component equations: x(t ) vox t 1 2 ax t 2 y (t ) voy t 1 2 a y t 2 Physics 1D03 - Lecture 8 24 Quiz A stone is thrown from the top of a building with a θ angle above the horizontal. In what direction (if any) does the stone experience an acceleration? A) Horizontal direction only B) Vertical direction only C) Both, horizontal and vertical directions D) No acceleration once it leaves your hand Physics 1D03 - Lecture 8 25 Projectile Motion: The Equations If a is constant (magnitude and direction), then: v (t ) vo a t 2 1 r (t ) vo t 2 a t In 2-D free-fall type questions, we separate the motion into horizontal and vertical parts: vx vox x vox t and v y voy gt y voy t 1 2 g t 2 < Horizontal, a = 0 < Vertical, a = -g Physics 1D03 - Lecture 8 26 Quiz Tarzan has a new slingshot. George the gorilla hangs from a tree, and bets that Tarzan can’t hit him. Tarzan aims at George, and is sorry that he didn’t pay more attention in physics class. Why ? (neglect air resistance) A) The stone will go over the gorilla’s head B) The stone will go below the gorilla C) The earth’s rotation must be taken into account Physics 1D03 - Lecture 8 27 Projectile Motion y=voyt-½gt2 x=voxt y vo x Physics 1D03 - Lecture 8 28 Example A stone is thrown upwards from the top of a 45.0 m high building with a 30º angle above the horizontal. If the initial velocity of the stone is 20.0 m/s, how long is the stone in the air, and how far from the base of the building does it land ? Physics 1D03 - Lecture 8 29 Height, Range, Time Determine the maximum height of a projectile Determine how far it goes (range) Determine the total time in flight Physics 1D03 - Lecture 8 30 Problem A stone is thrown from the top of a building with a θ angle above the horizontal with an initial velocity vo. How high will it go? How far will it go? How long does is it in the air for? Physics 1D03 - Lecture 8 31