Lecture 9 Ch6. Friction and Centripetal Force University Physics: Mechanics Dr.-Ing. Erwin Sitompul http://zitompul.wordpress.com Homework 7: Coin On A Book The figure below shows a coin of mass m at rest on a book that has been tilted at an angle θ with the horizontal. By experimenting, you find that when θ is increased to 13°, the coin is on the verge of sliding down the book, which means that even a slight increase beyond 13° produces sliding. What is the coefficient of static friction μs between the coin and the book? Hint: Draw the free-body diagram of the coin first. Erwin Sitompul University Physics: Mechanics 9/2 Solution of Homework 7: Coin On A Book Forces along the y axis: Fnet,y m a y F N Fg cos 0 • Why zero? FN Fg cos FN m g cos Forces along the x axis: Fnet,x m a x F g sin f s 0 • Why zero? F g sin s F N 0 m g sin s m g cos 0 s sin cos tan h d So, the coefficient of static friction is: s tan 13 0.231 Erwin Sitompul University Physics: Mechanics 9/3 Example: Blue Block → A block of mass m = 3 kg slides along a floor while a force F of magnitude 12 N is applied to it at an upward angle θ. The coefficient of kinetic friction between the block and the floor is μk = 0.4. We can vary θ from 0 to 90° (with the block remains on the floor. What θ gives the maximum value of the block’s acceleration magnitude a? Erwin Sitompul University Physics: Mechanics 9/4 Example: Blue Block Forces along the y axis: Forces along the x axis: Fnet,y m a y F N F y Fg 0 Fnet,x m a x Fx f k m a FN m g F sin F cos k FN m a F a cos k g sin m m F Erwin Sitompul • What θ gives the maximum value of a? • da/dθ = 0 University Physics: Mechanics 9/5 Example: Blue Block If a is given by F a cos k g sin m m F then, the derivative of a with respect to θ is da d F m sin k F cos 0 m tan k tan 1 k 1 tan (0.4) 21.80 Erwin Sitompul University Physics: Mechanics 9/6 Example: Two Blocks Block B in the figure below weighs 711 N. The coefficient of static friction between block and table is 0.25; angle θ is 30°. Assume that the cord between B and the knot is horizontal. Find the maximum weight of block A for which the system will be stationary. Erwin Sitompul University Physics: Mechanics 9/7 Example: Two Blocks → TW → TB Block B → TB → FgB Knot → TW → TA → TA → FNB → fs,max Wall Block A → FgA → TW → fs,max Knot → FgA Erwin Sitompul University Physics: Mechanics 9/8 Example: Two Blocks Forces along the y axis: → TW TWy Fnet,y 0 T W y FgA 0 θ → fs,max TW sin m A g Knot TWx Forces along the x axis: Fnet,x 0 mA g T W x f s,m ax 0 sin TW cos s FN B TW cos s m B g WA sin smB g cos sW B → FgA cos W A sW B tan (0.25)(711) tan 30 102.624 N Erwin Sitompul University Physics: Mechanics 9/9 Example: Multiple Objects A block of mass m1 on a rough, horizontal surface is connected to a ball of mass m2 by a lightweight cord over a lightweight, frictionless pulley as shown in the figure below. A force of magnitude F at an angle θ with the horizontal is applied to the block as shown and the block slides to the right. The coefficient of kinetic friction between the block and surface is μk. Find the magnitude of acceleration of the two objects. Erwin Sitompul University Physics: Mechanics 9/10 Example: Multiple Objects → FN Fy → fk m1 → Fg1 → T F θ → T → Fx Forces in m1 Fnet,x m1 a1 x F x T f k m1 a Forces in m2 m2 Fnet,y m 2 a 2 y T Fg2 m 2 a → Fg2 T m2 ( g a ) F cos T k FN m1 a T F cos m1 a k FN Fnet,y 0 F y F N Fg1 0 FN Fg1 F y FN m1 g F sin Erwin Sitompul University Physics: Mechanics 9/11 Example: Multiple Objects T m2 ( g a ) T F cos m1 a k FN FN m1 g F sin m 2 ( g a ) F cos m1 a k ( m1 g F sin ) m1 a m 2 a F cos k F sin k m1 g m 2 g ( m1 m 2 ) a F (cos k sin ) ( k m1 m 2 ) g a Erwin Sitompul F (cos k sin ) ( k m 1 m 2 ) g m1 m 2 University Physics: Mechanics 9/12 Example: Trio Blocks When the three blocks in the figure below are released from rest, they accelerate with a magnitude of 0.5 m/s2. Block 1 has mass M, block 2 has 2M, and block 3 has 2M. What is the coefficient of kinetic friction between block 2 and the table? Erwin Sitompul University Physics: Mechanics 9/13 Example: Trio Blocks a Forces in m1 a a Fnet,y m1 a1 y T1 Fg1 M a T1 M ( g a ) → T1 m1 → fk → Fg1 Erwin Sitompul → T2 → T2 m2 Fnet,x m 2 a 2 x T2 T1 f k 2 M a T2 T1 k FN 2 M a Fnet,y m 2 a 2 y → FN → T1 Forces in m2 m3 F N Fg2 0 FN 2 M g Forces in m3 Fnet,y m 3 a 3 y T2 Fg3 2 M ( a ) → Fg2 → Fg3 T2 2 M ( g a ) University Physics: Mechanics 9/14 Example: Trio Blocks T1 M ( g a ) T2 T1 k FN 2 M a FN 2 M g T2 2 M ( g a ) 2M (g a) M (g a) k 2Mg 2Ma k 2Mg 2M (g a) M (g a) 2Ma k k 2M (g a) M (g a) 2Ma 2Mg M g 5M a 2Mg g 5a 2g (9.8) 5(0.5) 2(9.8) 0 .3 7 2 m s Erwin Sitompul 2 University Physics: Mechanics 9/15