Physics 7C lecture 04 Dynamics Tuesday October 8, 8:00 AM – 9:20 AM Engineering Hall 1200

Two common free-body diagram errors The normal force must be perpendicular to the surface. There is no “ma force.” See Figure 5.13. 

A note on free-body diagrams Only the force of gravity acts on the falling apple. ma does not belong in a free-body diagram. 

When released, the cart accelerates up the ramp. Q5.3 A cart (weight w1) is attached by a lightweight cable to a bucket (weight w2) as shown. The ramp is frictionless. When released, the cart accelerates up the ramp. Which of the following is a correct free-body diagram for the cart? n n n n T T T T Answer: A m1a m1a w1 w1 w1 w1 A. B. C. D.

A5.3 A cart (weight w1) is attached by a lightweight cable to a bucket (weight w2) as shown. The ramp is frictionless. When released, the cart accelerates up the ramp. Which of the following is a correct free-body diagram for the cart? n n n n T T T T m1a m1a w1 w1 w1 w1 A. B. C. D.

Two bodies with the same acceleration We can treat the milk carton and tray as separate bodies, or we can treat them as a single composite body. Follow Example 5.11.

Q5.7 You are pushing a 1.00-kg food tray through the cafeteria line with a constant 9.0-N force. As the tray moves, it pushes on a 0.50-kg milk carton. If the food tray and milk carton move at constant speed, A. the tray exerts more force on the milk carton than the milk carton exerts on the tray. B. the tray exerts less force on the milk carton than the milk carton exerts on the tray. C. the tray exerts as much force on the milk carton as the milk carton exerts on the tray. Answer: C

A5.7 You are pushing a 1.00-kg food tray through the cafeteria line with a constant 9.0-N force. As the tray moves, it pushes on a 0.50-kg milk carton. If the food tray and milk carton move at constant speed, A. the tray exerts more force on the milk carton than the milk carton exerts on the tray. B. the tray exerts less force on the milk carton than the milk carton exerts on the tray. C. the tray exerts as much force on the milk carton as the milk carton exerts on the tray.

Q5.8 You are pushing a 1.00-kg food tray through the cafeteria line with a constant 9.0-N force. As the tray moves, it pushes on a 0.50-kg milk carton. If the food tray and milk carton are accelerating to the left, A. the tray exerts more force on the milk carton than the milk carton exerts on the tray. B. the tray exerts less force on the milk carton than the milk carton exerts on the tray. C. the tray exerts as much force on the milk carton as the milk carton exerts on the tray. Answer: C

A5.8 You are pushing a 1.00-kg food tray through the cafeteria line with a constant 9.0-N force. As the tray moves, it pushes on a 0.50-kg milk carton. If the food tray and milk carton are accelerating to the left, A. the tray exerts more force on the milk carton than the milk carton exerts on the tray. B. the tray exerts less force on the milk carton than the milk carton exerts on the tray. C. the tray exerts as much force on the milk carton as the milk carton exerts on the tray.

Two bodies with the same magnitude of acceleration The glider on the air track and the falling weight move in different directions, but their accelerations have the same magnitude. Follow Example 5.12 using Figure 5.15.

Bodies connected by a cable and pulley If the cart is moving at constant speed, what is w2/w1?

Bodies connected by a cable and pulley If the cart is moving at constant speed, what is w2/w1? consider the bucket T = w2

Bodies connected by a cable and pulley If the cart is moving at constant speed, what is w2/w1? consider the cart along x direction: W1 Sin 15o – T = 0 Thus: W2 = W1 Sin 15o and: W2/W1 = Sin 15o = 0.26

Straight-line motion with constant force The wind exerts a constant horizontal force on the boat. Follow Example 5.6.

Straight-line motion with friction For the ice boat in the previous example, a constant horizontal friction force now opposes its motion. Follow Example 5.7.

Tension in an elevator cable The elevator is moving downward but slowing to a stop. What is the tension in the supporting cable? Follow Example 5.8.

Apparent weight in an accelerating elevator A woman inside the elevator of the previous example is standing on a scale. How will the acceleration of the elevator affect the scale reading? Follow Example 5.9.

Acceleration down a hill What is the acceleration of a toboggan sliding down a friction-free slope? Follow Example 5.10.

Acceleration down a hill a = g sin α

D. not enough information given to decide Q5.6 A toboggan of weight w (including the passengers) slides down a hill of angle  at a constant speed. Which statement about the normal force on the toboggan (magnitude n) is correct? A. n = w B. n > w C. n < w D. not enough information given to decide Answer: C

A5.6 A toboggan of weight w (including the passengers) slides down a hill of angle  at a constant speed. Which statement about the normal force on the toboggan (magnitude n) is correct? A. n = w B. n > w C. n < w D. not enough information given to decide

B. the force of crate B on crate A C. the force F that you exert Q5.5 A lightweight crate (A) and a heavy crate (B) are side by side on a frictionless horizontal surface. You are applying a horizontal force F to crate A. Which of the following forces should be included in a free-body diagram for crate B? A. the weight of crate B B. the force of crate B on crate A C. the force F that you exert D. the acceleration of crate B E. more than one of the above F B A Answer: A

A5.5 A lightweight crate (A) and a heavy crate (B) are side by side on a frictionless horizontal surface. You are applying a horizontal force F to crate A. Which of the following forces should be included in a free-body diagram for crate B? A. the weight of crate B B. the force of crate B on crate A C. the force F that you exert D. the acceleration of crate B E. more than one of the above F B A

Friction What’s keeping geico on the wall?

Frictional forces When a body rests or slides on a surface, the friction force is parallel to the surface. Friction between two surfaces arises from interactions between molecules on the surfaces.

Kinetic and static friction Kinetic friction acts when a body slides over a surface. The kinetic friction force is fk = µkn. Static friction acts when there is no relative motion between bodies. The static friction force can vary between zero and its maximum value: fs ≤ µsn.

Static friction followed by kinetic friction Before the box slides, static friction acts. But once it starts to slide, kinetic friction acts.

Some approximate coefficients of friction

Friction in horizontal motion Before the crate moves, static friction acts on it. After it starts to move, kinetic friction acts. Follow Example 5.13.

Static friction can be less than the maximum Static friction only has its maximum value just before the box “breaks loose” and starts to slide. Follow Example 5.14.

A. block B accelerating to the right. Q5.9 Blocks A and C are connected by a string as shown. When released, block A accelerates to the right and block C accelerates downward. There is friction between blocks A and B, but not enough to prevent block B from slipping. If you stood next to the table during the time that block B is slipping on top of block A, you would see A. block B accelerating to the right. B. block B accelerating to the left. C. block B moving at constant speed to the right. D. block B moving at constant speed to the left. Answer: A

A5.9 Blocks A and C are connected by a string as shown. When released, block A accelerates to the right and block C accelerates downward. There is friction between blocks A and B, but not enough to prevent block B from slipping. If you stood next to the table during the time that block B is slipping on top of block A, you would see A. block B accelerating to the right. B. block B accelerating to the left. C. block B moving at constant speed to the right. D. block B moving at constant speed to the left.

Pulling a crate at an angle The angle of the pull affects the normal force, which in turn affects the friction force. Follow Example 5.15.

Pulling a crate at an angle The crate is moving right with acceleration of a = 1 m/s2, how big is T?

Pulling a crate at an angle vertical: n + T sin 30 = w = m g (1) horizontal: T cos 30 – fk = m a (2) and : fk = 0.4 n (3) 0.4 (1) + (2) T (0.4 sin 30+ cos 30) = 0.4 m g + m a T = (0.4 m g + m a) / (0.4 sin 30+ cos 30)

Motion on a slope having friction Consider the toboggan from Example 5.10, but with friction. Follow Example 5.16 and Figure 5.22. Consider the toboggan on a steeper hill, so it is now accelerating. Follow Example 5.17 and Figure 5.23.

A. the force of kinetic friction that the floor exerts on your shoes Q5.10 You are walking on a level floor. You are getting good traction, so the soles of your shoes don’t slip on the floor. Which of the following forces should be included in a free-body diagram for your body? A. the force of kinetic friction that the floor exerts on your shoes B. the force of static friction that the floor exerts on your shoes C. the force of kinetic friction that your shoes exert on the floor D. the force of static friction that your shoes exert on the floor E. more than one of these Answer: B

A5.10 You are walking on a level floor. You are getting good traction, so the soles of your shoes don’t slip on the floor. Which of the following forces should be included in a free-body diagram for your body? A. the force of kinetic friction that the floor exerts on your shoes B. the force of static friction that the floor exerts on your shoes C. the force of kinetic friction that your shoes exert on the floor D. the force of static friction that your shoes exert on the floor E. more than one of these

Fluid resistance and terminal speed The fluid resistance on a body depends on the speed of the body. A falling body reaches its terminal speed when the resisting force equals the weight of the body. The figures at the right illustrate the effects of air drag. Follow Example 5.18.