Concept #1: Conservative Forces are Path Independent

Practice: A 3 kg block is initially moving on a flat surface when it reaches the bottom of an inclined plane with 20 m/s. If the plane is smooth and makes 53° with the horizontal, how far up the plane will the block slide?

Example #1: Energy Problems in Inclines

Example #2: Energy Problems in Inclines

Example #3: Energy Problems in Inclines

Practice: When a 4-kg block is released from rest from the top of an inclined plane, it reaches the bottom with 4 m/s. The incline is 5 m long and makes 37° with the horizontal. Calculate the magnitude of the frictional force acting on the block.

Practice: When a block of unknown mass is released from the top of an inclined plane of length L meters, it slides down. The incline makes an angle of Θ degrees with the horizontal, and the coefficient of kinetic friction between the block and the plane is µ. Derive an expression for the speed of the block at the bottom of the plane.

Practice: A block of unknown mass is released from a distance D1 from the bottom of an inclined plane, then slides on a horizontal surface, and up a second inclined plane, as shown. Both planes make an angle of Θ degrees with the horizontal. The horizontal surface is smooth, but the coefficient of friction between the block and the two inclines is µ. Derive an expression for the maximum distance D2 that the block will reach on the second incline.

A box of mass m = 1.80 kg is at point A, which is at the top of an inclined plane of length ℓ = 2.40 m and inclination angle θ = 36.9°. The inclined angle is greased so that it is frictionless. The box slides down the inclined plane to the point B, where it starts to move horizontally across a surface with coefficient of kinetic friction μk. The box moves a distance d = 4.80 m across this surface before coming to rest at point C.
Find the gravitational potential energy of the box at point A (the top of the incline).

A long ramp made of cast iron is sloped at a constant
angle
kg = 52.0m/s
above the horizontal. Small blocks, each
with mass 0.42
m kg
but made of different materials, are released
from rest at a vertical height h above the bottom of the ramp. In
each case the coefficient of static friction is small enough that the
blocks start to slide down the ramp as soon as they are released.
You are asked to find h so that each block will have a speed of
4.00
m m/s
when it reaches the bottom of the ramp. You are given
these coefficients of sliding (kinetic) friction for different pairs of
materials
.Use work and energy considerations to find the required value
of h if the block is made from cast iron.Use work and energy considerations to find the required value
of h if the block is made from copper.Use work and energy considerations to find the required value
of h if the block is made from zinc.What is the required value of h for the copper block if its mass
is doubled to 0.84 kg?For a given block, if heta is increased while
h is kept the same, does the speed v of the block at the bottom of
the ramp increase, decrease, or stay the same?

A
12.0-kg
microwave oven is pushed
8.00 m
up the sloping surface of a loading ramp inclined at an angle of
36.1
above the horizontal, by a constant force vec{F} with a magnitude
113 N
and acting parallel to the ramp. The coefficient of kinetic friction between the oven and the ramp is
0.259.What is the work done on the oven by the force ?What is the work done on the oven by the friction force?Compute the increase in potential energy for the oven.Use your answers to parts A, B, and C to calculate the increase in the ovens kinetic energy.Use = m to calculate the ovens acceleration.Assuming that the oven is initially at rest, use the acceleration to calculate the ovens speed after the oven has traveled distance
8.00 m .From the data of the previous part, compute the increase in the ovens kinetic energy.

Truck brakes can fail if they get too hot. In some mountainous areas, ramps of loose gravel are constructed to stop runaway trucks that have lost their brakes. The combination of a slight upward slope and a large coefficient of rolling friction as the truck tires sink into the gravel brings the truck safely to a halt. Suppose a gravel ramp slopes upward at 6.0 and the coefficient of rolling friction is 0.36.Use work and energy to find the length of a ramp that will stop a 15,000 kg
truck that enters the ramp at 36 m/s .

A 16.0 kg child descends a slide 2.30 m high and reaches the bottom with a speed of 1.60 m/s .How much thermal energy due to friction was generated in this process?

A ski starts from rest and slides down a 26 incline 85 m long.If the coefficient of friction is 0.085, what is the skis speed at the base of the incline?If the snow is level at the foot of the incline and has the same coefficient of friction, how far will the ski travel along the level? Use energy methods.

A skier traveling 9.3 m/s reaches the foot of a steady upward 18 incline and glides 12 m up along this slope before coming to rest.What was the average coefficient of friction?

Justin, with a mass of
45
kg, is going down an 8.0-m-high water
slide. He starts at rest, and his speed at the bottom is
12
m/s.How
much thermal energy is created by friction during his descent?

A 62 kg skier starts from rest at the top of a 1200-m-long trail which drops a total of 210 m from top to bottom. At the bottom, the skier is moving 13.0 m/s .How much energy was dissipated by friction?

While a roofer is working on a roof that slants at 41.0 above the horizontal, he accidentally nudges his 91.0 N toolbox, causing it to start sliding downward, starting from rest.If it starts 4.55 m from the lower edge of the roof, how fast will the toolbox be moving just as it reaches the edge of the roof if the kinetic friction force on it is 20.0 N ?

A 28-kg rock approaches the foot of a hill with a speed of 15 m/s. This hill slopes upward at a constant angle of 40.0 above the horizontal. The coefficients of static and kinetic friction between the hill and the rock are 0.75 and 0.20, respectively.Use energy conservation to find the maximum height above the foot of the hill reached by the rock.Will the rock remain at rest at its highest point, or will it slide back down the hill?If the rock does slide back down, find its speed when it returns to the bottom of the hill.

A 1560 kg rocket is to be launched with an initial upward speed of 53.0 m/s . In order to assist its engines, the engineers will start it from rest on a ramp that rises 53 above the horizontal (the figure ). At the bottom, the ramp turns upward and launches the rocket vertically. The engines provide a constant forward thrust of 2000 N, and friction with the ramp surface is a constant 500 N.How far from the base of the ramp should the rocket start, as measured along the surface of the ramp?

A 61.0-kg skier starts from rest at the top of a ski slope of height 66.0 m .If frictional forces do −1.06×104 J of work on her as she descends, how fast is she going at the bottom of the slope?Now moving horizontally, the skier crosses a patch of soft snow, where the coefficient of friction is 0.21. If the patch is of width 66.0 m and the average force of air resistance on the skier is 170 N , how fast is she going after crossing the patch?After crossing the patch of soft snow, the skier hits a snowdrift and penetrates a distance 2.6 m into it before coming to a stop. What is the average force exerted on her by the snowdrift as it stops her?

A truck with mass m has a brake failure while going down an icy mountain road of constant downward slope angle 0
. Initially the truck is moving downhill at speed L. After careening downhill a distance r with negligible friction, the truck driver steers the runaway vehicle onto a runaway truck ramp of constant upward slope angle eta. The truck ramp has a soft sand surface for which the coefficient of rolling friction is mu_r.What is the distance that the truck moves up the ramp before coming to a halt? Solve using energy methods.

What minimum speed does a 160 g puck need to make it to the top of a 4.60 m -long, 25.0 frictionless ramp?

Sam, whose mass is
71 kg
, straps on his skis and starts down a
56 m
-high, 20 20 N frictionless slope. A strong headwind exerts a horizontal force of 200
m N
on him as he skies.Use work and
energy to find Sams speed at the bottom.

A novice skier, starting from rest, slides down a frictionless 13.0 incline whose vertical height is 130 {
m {
m m}} .How fast is she going when she reaches the bottom?

A crate of mass M starts from rest at the top of a frictionless ramp inclined at an angle alpha above the horizontal. Find its speed at the bottom of the ramp, a distance exttip{d}{d} from where it started. Do this in two ways.Take the level at which the potential energy is zero to be at the bottom of the ramp with exttip{y}{y} positive upward.Take the zero level for potential energy to be at the top of the ramp with exttip{y}{y} positive upward.Why did the normal force not enter into your solution?

You are designing a ski jump ramp for the next Winter Olympics. You need to calculate the vertical height h from the starting gate to the bottom of the ramp. The skiers push off hard with their ski poles at the start, just above the starting gate, so they typically have a speed of 2.0 m/s as they reach the gate. For safety, the skiers should have a speed of no more than 30.0 m/s when they reach the bottom of the ramp. You determine that for a 83.0 kg skier with good form, friction and air resistance will do total work of magnitude 4000 J on him during his run down the slope. What is the maximum height h for which the maximum safe speed will not be exceeded?

A box of mass m = 1.80 kg is at point A, which is at the top of an inclined plane of length ℓ = 2.40 m and inclination angle θ = 36.9°. The inclined angle is greased so that it is frictionless. The box slides down the inclined plane to the point B, where it starts to move horizontally across a surface with coefficient of kinetic friction μk. The box moves a distance d = 4.80 m across this surface before coming to rest at point C.Calculate μk. Hint: if there were no friction, the box would not stop moving. What does this tell you about the work done against friction over the distance d?

A block of mass 10.0 kg slides 16.0 m down a 36.9° incline, from point A at the top of the incline to point B at the bottom. As the block moves from point A to point B, the surface of the incline exerts a constant friction force that has magnitude 42.0 N.If the block has an initial speed of 8.0 m/s at point A, what is the speed of the block when it reaches point B?

A sled is initially given a shove up a frictionless 23.0° incline. It reaches a maximum vertical height 1.27 m {
m {
m m}}higher than where it started. What was its initial speed?

What height does a frictionless playground slide need so that a 35 kg child reaches the bottom at a speed of 5.7 m/s?

A 5.00-kg block is set into motion up an inclined plane with an initial speed of vi = 8.00 m/s (Fig. P8.23). The block comes to rest after traveling d = 3.00 m along the plane, which is inclined at an angle of θ = 30.08 to the horizontal. For this motion, determine(a) the change in the block’s kinetic energy,(b) the change in the potential energy of the block– Earth system, and(c) the friction force exerted on the block (assumed to be constant).(d) What is the coefficient of kinetic friction?