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## Chapter 7: Forces and Motion in 2D

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**Chapter 7: Forces and Motion in 2D**Dr. Zalesinsky www.YourJediMaster.com**Parts of the Chapter**7.1 Forces n 2D Equilibrium and Inclined Planes Independence of 2D Motion, Projectiles Horizontally Launched and Launched at an Angle 7.2 Projectile Motion 7.3 Circular Motion Centripetal Acceleration, Uniform Circular Motion, and Torque**7.1**Forces in 2D**4.11 Equilibrium Application of Newton’s Laws of Motion**Definition of Equilibrium An object is in equilibrium when it has zero acceleration.**4.11 Equilibrium Application of Newton’s Laws of Motion**• Reasoning Strategy • Select an object(s) to which the equations of equilibrium are • to be applied. • Draw a free-body diagram for each object chosen above. • Include only forces acting on the object, not forces the object • exerts on its environment. • Choose a set of x, y axes for each object and • resolve all forces • in the free-body diagram into components that point • along these • axes. • Apply the equations and solve for the • unknown quantities.**4.11 Equilibrium Application of Newton’s Laws of Motion**The first equation gives Substitution into the second gives**4.12 Nonequilibrium Application of Newton’s Laws of Motion**When an object is accelerating, it is not in equilibrium.**4.12 Nonequilibrium Application of Newton’s Laws of Motion**The acceleration is along the x axis so**4.11.1. Consider the following: (i) the book is at rest,**(ii) the book is moving at a constant velocity, (iii) the book is moving with a constant acceleration. Under which of these conditions is the book in equilibrium? a) (i) only b) (ii) only c) (iii) only d) (i) and (ii) only e) (ii) and (iii) only**4.11.1. Consider the following: (i) the book is at rest,**(ii) the book is moving at a constant velocity, (iii) the book is moving with a constant acceleration. Under which of these conditions is the book in equilibrium? a) (i) only b) (ii) only c) (iii) only d) (i) and (ii) only e) (ii) and (iii) only**4.11.2. A block of mass M is hung by ropes as shown. The**system is in equilibrium. The point O represents the knot, the junction of the three ropes. Which of the following statements is true concerning the magnitudes of the three forces in equilibrium? a) F1 + F2 = F3 b) F1 = F2 = 0.5×F3 c) F1 = F2 = F3 d) F1 > F3 e) F2 < F3**4.11.2. A block of mass M is hung by ropes as shown. The**system is in equilibrium. The point O represents the knot, the junction of the three ropes. Which of the following statements is true concerning the magnitudes of the three forces in equilibrium? a) F1 + F2 = F3 b) F1 = F2 = 0.5×F3 c) F1 = F2 = F3 d) F1 > F3 e) F2 < F3**4.11.3. A team of dogs pulls a sled of mass 2m with a force**. A second sled of mass m is attached by a rope and pulled behind the first sled. The tension in the rope is . Assuming frictional forces are too small to consider, determine the ratio of the magnitudes of the forces and , that is, P/T. a) 3 b) 2 c) 1 d) 0.5 e) 0.33**4.11.3. A team of dogs pulls a sled of mass 2m with a force**. A second sled of mass m is attached by a rope and pulled behind the first sled. The tension in the rope is . Assuming frictional forces are too small to consider, determine the ratio of the magnitudes of the forces and , that is, P/T. a) 3 b) 2 c) 1 d) 0.5 e) 0.33**Motion along an Inclined Plane**See pp. 152 – 154 in text**7.2**Projectile motion**3.3 Projectile Motion**Under the influence of gravity alone, an object near the surface of the Earth will accelerate downwards at 9.80m/s2.**3.3 Projectile Motion**Example 3 A Falling Care Package The airplane is moving horizontally with a constant velocity of +115 m/s at an altitude of 1050m. Determine the time required for the care package to hit the ground.**3.3 Projectile Motion**Example 4 The Velocity of the Care Package What are the magnitude and direction of the final velocity of the care package?**3.3 Projectile Motion**Conceptual Example 5 I Shot a Bullet into the Air... Suppose you are driving a convertible with the top down. The car is moving to the right at constant velocity. You point a rifle straight up into the air and fire it. In the absence of air resistance, where would the bullet land – behind you, ahead of you, or in the barrel of the rifle?**3.3 Projectile Motion**Example 6 The Height of a Kickoff A placekicker kicks a football at and angle of 40.0 degrees and the initial speed of the ball is 22 m/s. Ignoring air resistance, determine the maximum height that the ball attains.**3.3 Projectile Motion**Example 7 The Time of Flight of a Kickoff What is the time of flight between kickoff and landing?**3.3 Projectile Motion**Example 8 The Range of a Kickoff Calculate the range R of the projectile.**3.3 Projectile Motion**Conceptual Example 10 Two Ways to Throw a Stone From the top of a cliff, a person throws two stones. The stones have identical initial speeds, but stone 1 is thrown downward at some angle above the horizontal and stone 2 is thrown at the same angle below the horizontal. Neglecting air resistance, which stone, if either, strikes the water with greater velocity?**3.3.1. A football is kicked at an angle 25 with respect**to the horizontal. Which one of the following statements best describes the acceleration of the football during this event if air resistance is neglected? a) The acceleration is zero m/s2 at all times. b) The acceleration is zero m/s2 when the football has reached the highest point in its trajectory. c) The acceleration is positive as the football rises, and it is negative as the football falls. d) The acceleration starts at 9.8 m/s2 and drops to some constant lower value as the ball approaches the ground. e) The acceleration is 9.8 m/s2 at all times.**3.3.1. A football is kicked at an angle 25 with respect**to the horizontal. Which one of the following statements best describes the acceleration of the football during this event if air resistance is neglected? a) The acceleration is zero m/s2 at all times. b) The acceleration is zero m/s2 when the football has reached the highest point in its trajectory. c) The acceleration is positive as the football rises, and it is negative as the football falls. d) The acceleration starts at 9.8 m/s2 and drops to some constant lower value as the ball approaches the ground. e) The acceleration is 9.8 m/s2 at all times.**3.3.2. A baseball is hit upward and travels along a**parabolic arc before it strikes the ground. Which one of the following statements is necessarily true? a) The velocity of the ball is a maximum when the ball is at the highest point in the arc. b) The x-component of the velocity of the ball is the same throughout the ball's flight. c) The acceleration of the ball decreases as the ball moves upward. d) The velocity of the ball is zero m/s when the ball is at the highest point in the arc. e) The acceleration of the ball is zero m/s2 when the ball is at the highest point in the arc.**3.3.2. A baseball is hit upward and travels along a**parabolic arc before it strikes the ground. Which one of the following statements is necessarily true? a) The velocity of the ball is a maximum when the ball is at the highest point in the arc. b) The x-component of the velocity of the ball is the same throughout the ball's flight. c) The acceleration of the ball decreases as the ball moves upward. d) The velocity of the ball is zero m/s when the ball is at the highest point in the arc. e) The acceleration of the ball is zero m/s2 when the ball is at the highest point in the arc.**3.3.3. Two cannons are mounted on a high cliff. Cannon A**fires balls with twice the initial velocity of cannon B. Both cannons are aimed horizontally and fired. How does the horizontal range of cannon A compare to that of cannon B? a) The range for both balls will be the same b) The range of the cannon ball B is about 0.7 that of cannon ball A. c) The range of the cannon ball B is about 1.4 times that of cannon ball A. d) The range of the cannon ball B is about 2 times that of cannon ball A. e) The range of the cannon ball B is about 0.5 that of cannon ball A.**3.3.3. Two cannons are mounted on a high cliff. Cannon A**fires balls with twice the initial velocity of cannon B. Both cannons are aimed horizontally and fired. How does the horizontal range of cannon A compare to that of cannon B? a) The range for both balls will be the same b) The range of the cannon ball B is about 0.7 that of cannon ball A. c) The range of the cannon ball B is about 1.4 times that of cannon ball A. d) The range of the cannon ball B is about 2 times that of cannon ball A. e) The range of the cannon ball B is about 0.5 that of cannon ball A.**3.3.4. Which one of the following statements concerning the**range of a football is true if the football is kicked at an angle with an initial speed v0? a) The range is independent of initial speed v0. b) The range is only dependent on the initial speed v0. c) The range is independent of the angle. d) The range is only dependent on the angle. e) The range is dependent on both the initial speed v0 and the angle.**3.3.4. Which one of the following statements concerning the**range of a football is true if the football is kicked at an angle with an initial speed v0? a) The range is independent of initial speed v0. b) The range is only dependent on the initial speed v0. c) The range is independent of the angle. d) The range is only dependent on the angle. e) The range is dependent on both the initial speed v0 and the angle.**3.3.5. A bullet is aimed at a target on the wall a distance**L away from the firing position. Because of gravity, the bullet strikes the wall a distance Δybelow the mark as suggested in the figure. Note: The drawing is not to scale. If the distance L was half as large, and the bullet had the same initial velocity, how would Δybe affected? a) Δy will double. b) Δywill be half as large. c) Δywill be one fourth as large. d) Δywill be four times larger. e) It is not possible to determine unless numerical values are given for the distances.**3.3.1. A bicyclist is riding at a constant speed along a**horizontal, straight-line path. The rider throws a ball straight up to a height a few meters above her head. Ignoring air resistance, where will the ball land? a) in front of the rider b) behind the rider c) in the same hand that threw the ball d) in the opposite hand to the one that threw it e) This cannot be determined without knowing the speed of the rider and the maximum height of the ball.