Unit 2: DynamicsFor this unit you must:1. Model verbally or visually the properties of a system based on its substructure and to relate this to

changes in the system properties over time as external variables are changed2. Design an experiment for collecting data to determine the relationship between the net force exerted

on an object, its inertial mass and its acceleration3. Design a plan for collecting data to measure gravitational mass and to measure inertial mass and to

distinguish between the two experiments4. Apply F = mg to calculate the gravitational force on an object with mass m in a gravitational field of

strength g in the context of the effects of a net force on objects and systems.5. Represent forces in diagrams or mathematically using appropriately labeled vectors with magnitude,

direction, and units during the analysis of a situation 6. Analyze a scenario and make claims (develop arguments, justify assertions) about the forces exerted

on an object by other objects for different types of forces or components of forces7. Challenge a claim that an object can exert a force on itself8. Describe a force as an interaction between two objects and identify both objects for any force9. Construct explanations of physical situations involving the interaction of bodies using Newton’s third

law and the representation of action-reaction pairs of forces10. Use Newton’s third law to make claims and predictions about the action-reaction pairs of forces when

two objects interact11. Analyze situations involving interactions among several objects by using free-body diagrams that

include application of Newton’s third law to identify forces12. Predict the motion of an object subject to forces exerted by several objects using an application of

Newton’s second law in a variety of physical situations with acceleration in one dimension13. Design a plan to collect and analyze data for motion (static, constant, or accelerating) from force

measurements and carry out analysis to determine the relationship between the net force and the vector sum of the individual forces

14. Re-express a free-body diagram representation into a mathematical representation and solve the mathematical representation for the acceleration of the object

15. Create and use free-body diagrams to analyze physical situations to solve problems with motion qualitatively and quantitatively

16. Make claims about various contact forces between objects based on the microscopic cause of those forces

17. Explain contact forces (tension, friction, normal, spring) as arising from interatomic electric forces and that they therefore have certain directions

18. Use representations of the center of mass of an isolated two-object system to analyze the motion of the system qualitatively and semiqualitatively

19. Evaluate using given data whether all the forces on a system or whether all the parts of a system have been identifies

20. Apply Newton’s second law to systems to calculate the change in the center-of-mass velocity when an external force is exerted on the system

21. Use visual or mathematical representations of the forces between objects in a system to predict whether or not there will be a change in the center-of-mass velocity of that system

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Chapter 5: Newton’s Laws of Motion

What is force? A force is exerted due to the interaction of that object with another object. An object cannot exert a force on itself. Force is a vector quantity. Contact forces are forces that act on an object by touching it Long-range forces are forces that act on an object without physical contact Objects at rest may have forces acting on it by other objects.

What is the difference between gravitational mass and inertial mass?

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Example 1: The metal head of a hammer is loose. To tighten it, you drop the hammer down onto a table. Should you (a) drop the hammer with the handle end down, (b) drop the hammer with the head end down, or (c) do you get the same result either way? Justify your response.

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Newton’s Second Law The acceleration of an object, but not necessarily its velocity, is always in the direction of the net force A coordinate system with one axis parallel to the direction of the acceleration simplifies the translation from the

free-body diagram to the algebraic representation.

Combining Newton’s Second Law with Kinematics Forces that systems exert on each other are due to interactions between object in the systems. If the

interacting objects are parts of the same system, there will be no change in the center-of-mass velocity of that system.

The linear motion of a system can be described by the displacement, velocity and acceleration of its center of mass.

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Example 2: Two boxes – one large and heavy, the other small and light – rest on a smooth, level floor. You push with a force F on either the small box or the large box. Is the contact force between the two boxes (a) the same in either case, (b) larger when you push on the large box, or (c) larger when you push on the small box? Justify your response.

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Example 3: A man is stranded on a raft with combined mass 1300 kg. By paddling he causes an average force of 17 N to be applied to the raft in a direction due east. The wind also exerts a force of 15 N [67°north of east]. What is the acceleration of the center of mass of the system?

Example 4: A Imperial Tie Fighter of mass 7500.0 kg travelling 80.0 km/h is under attack by a rebel X-Wing. The X-Wing’s lasers exert a force of 650.0 N as shown while the Tie Fighter’s engine thrusts at 1200.0 N. Find the velocity of the Tie Fighter after 15 s of the attack (assume Flaser remains at the same angle throughout the attack).

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Example 4: (continued)

Weight

A gravitational field g at the location of an object with mass m causes a gravitational force of magnitude mg to be exerted on the object in the direction of the field

Gravitational force = weight The gravitational field at a point in space is measured by dividing the gravitational force exerted by the field on a

test object at that point by the mass of the test object and has the same direction as the force If the gravitational force is the only force exerted on the object, the observed free-fall acceleration of the object

is equal to the magnitude of the gravitational field (N/kg) at that location

Normal Force Normal means perpendicular The force exerted by a surface against an object that is pressing against the surface

Example 5: Anjay’s mass is 70 kg. He is standing on a scale in an elevator that is moving at 5.0 m/s. As the elevator stops, the scale reads 750 N. Before it stopped, was the elevator moving up or down? How long did the elevator take to come to rest?

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Example 6: Can the normal force be greater than or less than the weight of an object? Give an example to prove.

Example 7: Example 7: A 6.0 kg block of ice is acted on by two forces, F1 and F2, as shown. If F1 = 13 N and F2 = 11 N, find (a) the acceleration of the ice and (b) the normal force exerted on it by the table.

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Chapter 6: Applications of Newton’s Laws

Contact forces result from the interaction of one object touching another object and they arise from interatomic electric forces. These forces include tension, friction and spring.

Friction No surface is smooth Friction depends on the surfaces in contact and on the normal force Friction is independent of the relative speed between the surfaces and their area of contact Kinetic friction is constant for objects in motion Static friction prevents an object from moving when a force is applied Static friction changes based on the applied force. It has a maximum value but may take on any value from zero

to maximum depending on what is needed to match the applied force A rolling wheel experiences friction but not kinetic friction. The portion of the wheel that contacts the surface is

stationary with respect to the surface, not sliding. We can treat this friction like another type of friction called rolling friction and calculate it using the same formula.

Example 8: A car traveling at 20 m/s stops in a distance of 50 m. Assume that the deceleration is constant. The coefficients of friction between a passenger and the seat are µs = 0.5 and µk = 0.03. Will a 70-kg passenger slide off the seat if not wearing a seat belt?

Example 9: An 18.0 kg mass is on a rough table with a coefficient of static friction of 0.45. A rope is attached at an angle of 50.0° to the horizontal. (a) Would a force of 100.0 N along the rope move the mass? (b) What is the minimum force, along the rope, required to move the mass?

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Example 9 (continued)

Example 10: A box of mass M is pulled along a horizontal surface with an applied force F at an angle ϴ above the horizontal. If the acceleration of the box is a, determine the expression for the coefficient of kinetic friction.

Example 11: A box of mass M slides across a floor and comes to a complete stop. If its initial speed was 10.0 km/h and the coefficient of kinetic friction is 0.34, find the distance the box travels before coming to rest.

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Example 12: A trained sea lion slides from rest down a 3.0 m long ramp into a pool of water. If the ramp is inclined at 23° above the horizontal and the coefficient of kinetic friction between the sea lion and the ramp is 0.26, how long does it take the sea lion to make a splash in the pool?

Strings and Springs

Tension Tension in a rope is an example of Newton’s third law If a rope is considered to have negligible mass the tension is equal throughout the length of the rope Ideal pulleys change the direction of tension without changing its magnitude (later we will deal with pulleys that

have friction) When solving problems with a system of masses attached by a string, treat the system as an object to find the

acceleration, and then solve for tension in the strings by isolating each object separately within the system

Example 13: Is the reading of the scale (a) greater than, (b) less than or (c) equal to 49 N? Justify your response

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Example 14: A rope of mass m is attached to a ceiling and holding a box of mass M at rest. As you move further up the rope the tension in the rope is: (a) less than Mg, (b) equal to Mg or (c) greater than Mg. Justify your response.

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Example 15: Two blocks, one 0.80 kg and the other 2.0 kg are connected by a massless string over a frictionless pulley. The coefficient of friction is 0.14 and the downward ramp angle is 60.0°.

(a) Determine the acceleration of the blocks.(b) Calculate the tension of the string.

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Example 16: The pulley device pictured below is used to pull a wagon load of gold bearing ore out of a mine. The barrel is filled with water and then released from rest so that as it falls the wagon is pulled up the incline. The mass of the wagon and its load is 90.0 kg. The mass of the barrel filled with water is 60.0 kg. The incline angle is 30.0° and the barrel falls h = 10.0 m. If the coefficient of friction experienced by the cart is 0.15 and the system is released from rest, what is the speed of the wagon when the water barrel had fallen 10.0 m?

Example 17: Block 1, of mass m1, is connected over an ideal (massless and frictionless) pulley to block 2, of mass m2 as shown. Assume that the blocks accelerate as shown with an acceleration of a, and that the coefficient of kinetic friction between block 2 and the plane is µ. Find the ratio of the masses m1 / m2.

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Hooke’s Law A spring will exert a force that is proportional to the amount it is stretched or compressed. The spring force is considered a restoring force since it attempts to restore the system to equilibrium Forces that hold atoms together are often modelled by Hooke’s Law (normal force, vibrations, wave motion) Hooke’s law is valid up to a point. If the force is too great the object will excessively stretch or break

Example 18: A 4.2 kg ball is on top of a 28.6 m high building where it is dropped onto a target located 1.2 m above the ground. The target rests on a spring which will stop the ball over a distance of 0.60 m. Determine the spring constant of the spring.

Example 19: A 1.8 kg cart is on a frictionless table and attached to a wall, to the left, by a spring with a spring constant of 250 N/m. The cart is also attached to a 4.5 kg hanging mass. Determine the amount of deformation of the spring.

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Translational Equilibrium When the net force acting on an object is zero the object is in translational equilibrium If the net force and the net torque acting on the object are both zero, we say the object is in static equilibrium

Example 20: Calculate the tension in the three ropes A, B, and C if the 5.0 kg mass is in static equilibrium.

Example 21: The system below is in static equilibrium. Determine the mass of the monkey. In the process, also calculate T1, T2, T3, T4, T5, T6, and W2 if W1 = 350 N.

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Connected Objects Forces exerted on the system result in the acceleration of each object being the same Forces exerted within the system do not contribute to the net acceleration To solve for forces within the system treat each object as a separate system Choose the direction of motion to be a positive acceleration

Example 22: FIGURE 5.26 shows a 5.0 kg block A being pushed with a 3.0 N force. In front of this block is a 10 kg block B; the two blocks move together. What force does block A exert on block B?

Example 23: Two boxes of 8.5 kg and 12.2 kg are touching each other on a surface where the coefficient of static friction is 0.30 and the coefficient of kinetic friction is 0.10. (a) Determine if a force of 75 N will cause the boxes to move, and (b) determine the acceleration of the boxes if they do move.

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Example 24: Two blocks of masses m1 and m2 are placed in contact with each other on a frictionless horizontal surface. If a constant horizontal force, F, is applied to m1, determine the expression for (a) the acceleration of the masses and (b) the force experienced by each mass because of the other mass.

Example 25: Two masses m1 and m2 are connected by a string over a pulley. Mass m1 slides without friction on a horizontal tabletop and mass m2 falls vertically downward. Both masses move with a constant acceleration of magnitude a. Is the tension in the string greater than, less than, or equal to the weight of block 2, m 2g? Justify your response.

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Example 26: Three blocks (2.0 kg, 3.0 kg, and 4.0 kg respectively) are in contact with each other on a surface with µ = 0.10. If a force of 28 N is applied to m1, determine:

(a) The acceleration of the blocks(b) The magnitude of the contact forces between the blocks(c) The resultant force on each block

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Example 26: (continued)

Example 27: A 2.0 kg block is placed on top of a 5.0 kg block. The coefficient of kinetic friction between the 5.0 kg block and the surface is 0.20. A horizontal force is applied to the 5.0 kg block.

(a) Draw a FBD for each block(b) Calculate the force necessary to pull both blocks to the right with an acceleration of 3.0 m/s2.(c) Find the minimum coefficient of static friction between the blocks such that the 2.0 kg block does not

slip under the acceleration of 3.0 m/s2.

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Example 28: A large cube of mass 25 kg is being accelerated across a horizontal frictionless surface by a horizontal force P. A small cube of mass 4.0 kg is in contact with the front surface of the large cube and will slide downward unless P is sufficiently large. The coefficient of static friction between the cubes is 0.71.

(a) What is the smallest magnitude that P can have in order to keep the small cube from sliding downwards?(b) Find the force of the small cube on the large cube? (c) Find P if the coefficient of friction between the floor and the 25 kg block is 0.20.

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P

Circular Motion

Example :

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