Inclined Plane Problems When the surface is not perpendicular to the direction of gravity, you should select a coordinate system that aligned with the plane’s surface θ
Components of Weight on an IP θ mg
Example 7-1 A child of mass m = 10.0 kg rides on a toboggan down a 20.0 m long slick, ice-covered (signal: friction is negligible) hill inclined at an angle of θ = 20° with respect to the horizontal. (a) What is the acceleration of the child? (b) What is the normal force exerted on the child by the toboggan?
Example 7-2 Two blocks are connects by a string as shown in the picture. The frictionless inclined surface makes an angle of 35° with the horizontal, and the block on the incline has a mass of 5.7 kg. (a) Find the mass of the hanging block m such that the system is in equilibrium. (b) Find the direction and the magnitude of the hanging block’s acceleration if the hanging mass has a mass m = 3.0 kg. (c) Find the direction and the magnitude of the hanging block’s acceleration if the hanging mass has a mass m = 4.2 kg.
Friction Friction is a force that opposes the relative motion of two surfaces past each other It is a consequence of microscopic hills and valleys “catching” on each other Smoother surface => smoother hills & valleys => less friction
Kinetic Friction When two surfaces move past each other (rub), kinetic friction opposes their motion It is independent of the relative speeds and the surface area of contact It is proportional to the normal force, the constant of proportionality is called the coefficient of kinetic friction
Coefficient of Friction The coefficient of friction is dimensionless and unitless It must be experimentally determined Depends on how to “roughness” of a surface interacts with the “roughness” of another surface
Example 7-3 If a car’s wheels are “locked” (kept from rolling) during emergency braking, the car slides along the road. Ripped-off bits of tire and small melted sections of the road form the “skid marks” that reveal distance that a car slides during the braking. The record for the longest skid marks on a public road was reportedly set by a Jaguar on the M1 highway in England. The marks were 290 m long! Assuming that μ k = 0.6 and the car’s acceleration was constant during the braking, how fast was the car going when the wheel’s became locked?
Example 7-4 A trained sea lion slides from rest with constant acceleration down a 3.0-m long ramp into a pool of water. If the ramp is inclined at and angle of 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 for the sea lion to make a splash in the pool?
Example 7-5 Billy pulls his 35 kg toybox with a rope along a horizontal floor with a constant velocity. The coefficient of kinetic friction between the box and the floor is 0.15 and the rope makes an angle of 25° with the horizontal. (a) What is the tension in the rope? (b) If Billy pulls on the rope harder, so that T is greater than in part a, is the magnitude of the force of kinetic friction greater than, less than, or the same as in part a? (c) Find the angle of the rope with the horizontal such that Billy can pull with the minimum force needed for the toybox to move with constant velocity.
Example 7-6 (Bailey’s FAV) A slide-loving pig slides down a certain 27° incline in twice the time it would take it to slide down the same slide if it were frictionless. Find the coefficient of kinetic friction for the slide.
Static Friction Static Friction is responsible for keeping two surfaces from moving relative to each other It responds to provide exactly enough force to keep the surfaces at rest relative to each other (like normal force) Has a maximum value (f s,max ) Is parallel to surfaces in contact in the direction that opposes relative motion
Example 7-7 A flatbed truck truck slowly tilts its bed upward to dispose of a 95.0-kg crate. For small angles of tilt the crate stays put, but when the tilt angle exceeds 23.2°, the crate begins to slide. What is the coefficient of static friction
Example 7-8 The two blocks (with m = 16 kg and M = 88 kg) shown in the picture are not attached. The coefficient of static friction between the blocks is μ s = 0.38, but the surface beneath the larger block is frictionless. What is the minimum magnitude of the horizontal force F required to keep the smaller block from slipping down the larger block?
Example 7-8 A 49-kg rock climber is performing a “chimney climb” between two vertical vertical rock slabs. The coefficient of static friction between her shoes and the rock is 1.2; between her back and the rock is 0.8. (a) What is the minimum force that she needs to push on the rock so that she does not slip? (b) What fraction of her weight is supported by her shoes when pushing with this minimum force?
Example 7-10 In the picture, blocks A and B have weights 44 N and 22 N, respectively. (a) Determine the minimum weight of block C to keep A from sliding if μ s between A and the table is (b) Block C suddenly is lifted off A. What is the acceleration of block A if μ k between A and the table is 0.15?