Lecture 5. Dynamics. Forces: Newton s First and Second

Transcription

1 Lecture 5 Dynamics. Forces: Newton s First and Second

2 What is a force? It s a pull or a push: F F Force is a quantitative description of the interaction between two physical bodies that causes them to undergo a change in speed, change in direction of motion, or a change of shape. It has a magnitude and a direction Forces are vectors!

3 Types of forces 1. Contact forces: - Normal force (always perpendicular to contact surface); it s the basic push - Tension (force through a string, cable, rope, etc): it s the basic pull - Friction (appears when two surfaces slide or try to slide one against the other) 2. Action at distance or non-contact forces: (The fundamental forces of Nature!) - Gravitational - Electric and Magnetic - Weak force (important in subatomic world) - Strong force (important in subatomic world)

4 Newton s Three Laws 1. Any object remains at rest or in motion along a straight line with constant speed unless acted upon by a net force. 2. Fnet = F = m a all 3. Forces occur in pairs: F on B by A = F on A by B For every force (or action) there is an equal but opposite force (or reaction)

5 Newton s First Law Any object remains at rest or in motion along a straight line with constant speed unless acted upon by a net force This law tells you what happens in the absence of any force, for instance in outer space DEMO: Air track

6 ACT: Snapped string A small ball attached to the end of a string moves in horizontal circles on a frictionless table as shown below. If the string snaps, what will be the trajectory of the ball?

7 ACT: Snapped string A small ball attached to the end of a string moves in horizontal circles on a frictionless table as shown below. If the string snaps, what will be the trajectory of the ball? A B C

8 Newton s Second Law The net force applied to an object is proportional to its acceleration: F net = m a Units: 1 N = 1 kg m/s 2 (1 lb = N) This is in fact the definition of mass! (The proportionality factor, m.) a a /2 m F 2m F DEMO: Toy truck

9 ACT: Force and acceleration A force F acting on a mass m 1 results in an acceleration a 1. The same force acting on a mass m 2 results in an acceleration a 2 = 2a 1. F a 1 m 1 F a 2 = 2a 1 m 2 If both masses are put together and the same force is applied to the combination, what is the resulting acceleration? A. 2/3 a 1 a? B. 3/2 a 1 F m 1 m 2 C. 3/4 a 1

10 ACT: Newton's First Law of Motion If a car is moving at a constant velocity (constant speed along a straight line): 1. A constant total force must be acting on it to keep it moving. 2. There must be no friction force acting on it. 3. The net force on it must be zero 4. There must be no force acting on it.

11 EXAMPLE: Pushing a box on ice. A skater is pushing a heavy box (m = 100 kg) across a sheet of ice (horizontal and frictionless). He applies a horizontal force of 50 N on the box. If the box starts at rest, what is its speed v after being pushed over a distance d = 10 m?

12 EXAMPLE: Pushing a box on ice. A skater is pushing a heavy box (m = 100 kg) across a sheet of ice (horizontal and frictionless). He applies a horizontal force of 50 N on the box. If the box starts at rest, what is its speed v after being pushed over a distance d = 10 m? a m = 100 kg F = 50 N v 0 = 0 d = 10 m v?

13 A skater is pushing a heavy box (m = 100 kg) across a sheet of ice (horizontal and frictionless). He applies a horizontal force of 50 N on the box. If the box starts at rest, what is its speed v after being pushed over a distance d = 10 m? a m = 100 kg F = 50 N v 0 = 0 d = 10 m v? v 2 v 0 2 = 2a Δ x = v = v F m Δ x 0 2a Δ x 2Fd = = m target 3.2 m/s a = F net m = F m

14 Remember: What is proportional to the acceleration is the NET force. EXAMPLE: Two people are pulling on a box initially at rest with forces F 1 and F 2. In which direction will the box move? (top view) F 1 F net a F 2 The acceleration of the box points in the direction of. Fnet = F 1 + F 2

15 EXAMPLE: Two people are pulling on a box with forces F 1 and F 2. F 1 θ 1 F net a y θ 2 x F 2 Let θ 1 = 20, θ 2 = 40, m box = 5.0 kg and a = 1.1 m/s 2. What is the magnitude of each force? F F net, x = ma net,y = 0 F cosq + F cosq = ma 2 equations F sinq - F sinq = unknowns

16 y F 1 θ 2 θ 1 F net θ 1 = 20 θ 2 = 40 m box = 5.0 kg a = 1.1 m/s 2 F 2 x F sinq - F sinq = F cosq + F cosq = ma F 2 = sin θ 2 ma tan θ 1 +cos θ 2 F 2 = F 1 sin θ 2 sin θ 1 = = = sin θ 1 sin θ 2 ( sin θ 1 (5.0 kg) (1.1 m/s 2 ) sin 40 tan 20 ma sinq2 F1 = F2 sinq + cos 40 sin θ 2 tan θ 1 + cos θ 2) ma tan θ 2 + cos θ 1 1 = 2.2 N = (5.0 kg) (1.1 ) m/s2 sin 20 + cos 20 tan 40 F 2 ( = 4.1 N sin θ 2 sin θ 1 cos θ 1 Checks: Units Limits: + cos θ 2) = ma θ 1 = θ 2 θ 1 = θ 2 = 0 m 0, a 0,

17 Weight The weight of an object is the force of attraction of gravity by Earth on the object (usually near the Earth). It is observed that, near the surface of the Earth: g 9.8 m/s2 W = m g direction: toward the center of the Earth m W m W Weight and mass are not the same thing!! The direction and magnitude of weight (vector) changes in different places. Mass (scalar) is always the same. (On the Moon, for instance, g = 1.67 m/s 2 ) m W Moon

18 EXAMPLE 1: Block on table A block is placed on a horizontal surface. What forces are acting on the block? N by table Fnet = 0 N = W W by Earth If weight was the only force, there would be a net force on the box pointing down an acceleration pointing down! There has to be another force to achieve Fnet = 0

19 EXAMPLE 2: Block on two tables A block is balanced in the space between two tables as shown below. What forces are acting on the block? F net = 0 N by left table N by right table W by Earth N R = N L = W 2 (½ with enough symmetry) DEMO: Nail bed If the block rests on 100 mini-tables, each table exerts a relatively small force: N each table = W 100 DEMO: Sharing the weight

20 Optional: Tension in String Tension: magnitude of the force acting across a cross-section of the rope/string/cable at a given position (it s the force you would measure if you cut the rope and grabbed the ends).

21 Optional: Tension in String We ll assume ideal (constant length), massless strings (i.e, mass much smaller than the rest of the masses in the system). Consider a segment with mass m of a rope with an acceleration a to the right. If we neglect gravity, the forces on the segment are: T 1 m T 2 a T 2 T 1 = ma T 2 > T 1 If m= 0, T 2 T 1 = 0 T 2 =T 1 (also, then the weight of the segment really is negligible)

22 Optional: Tension in String Massless string: The tension is the same throughout the string. It can only pull in the direction of its length. T T This makes our lives a lot easier (and it is a good approximation most of the time). Constant length string: All objects attached to it move together (same acceleration and velocity)

23 Pegs and pulleys Used to route forces to different direction. Ideal massless pulley or ideal smooth peg: changes the direction of the force without changing its magnitude. T T T T W

24 EXAMPLE: Elevator moving up A 200-kg elevator begins moving up with an acceleration of 3.0 m/s 2. Find the tension exerted by the cable.

25 EXAMPLE: Elevator moving up A 200-kg elevator begins moving up with an acceleration of 3.0 m/s 2. Find the tension exerted by the cable. Draw a figure and select axes. Newton s second law: Fnet = m a T - mg = ma ( ) T = m a + g = 2 2 = (200 kg)(9.8 m/s m/s ) = = 2560 N Checks: If a increases, T increases. For a = 0, T = mg + T mg a

26 Back to Free Fall If we neglect friction, only one force is acting which is the wight (mg) downward: Newton s second law: F net = m a mg = ma a = g =9.8 m/s 2 All falling bodies have the same acceleration of 9.8 m/s 2 downward because: The m in mg and the m in Newton s second law are the same (Equivalence of Gravitational and Inertial Mass. This is the basis of General Relativity!). Weight is the only force acting! NOTE: Weight will always be there in problems near the Earth-, but most of the time, it is NOT the only force. So the acceleration will NOT be 9.8 m/s 2 for every problem. mg