As shown in the figure below a 225 kg block is released from...

As shown in the figure below, a 2.25 kg block is released from rest on a ramp of height h. An illustration shows a block positioned at a height h at the top of the vertical portion of a curved ramp. The ramp is oriented such that it curves downward from right to left, starting from an almost-vertical slope and then decreasing in slope magnitude until it is horizontal. The ramp is frictionless except for a horizontal 15.0 cm section at the bottom of the curve. When the block is released, it slides without friction to the bottom of the ramp, and then continues across a surface that is frictionless except for a rough patch of width 15.0 cm that has a coefficient of kinetic friction μk = 0.710. Find h (in m) such that the block's speed after crossing the rough patch is 3.50 m/s. (Enter a number.)

Transcript

00:01 In this problem, we're told that a block with a mass of 1 .75 kilograms travels down a frictionless ramp and passes over a rough patch of ground with a coefficient of kinetic friction of 0 .640.

00:16 The length of that patch is 0 .10 meters.

00:21 We're also given that the final velocity after passing over that patch and to the other side of it is, 3 .50 meters per second.

00:35 Given this information, we need to determine what the height of the ramp is.

00:40 So first we need to consider that all of the non -conservative work done, which is only friction in this case, is equal to the change in energy from the start of the friction patch to the end of the friction patch.

00:59 So i'll say final energy, minus initial energy.

01:08 Now, we know what the final energy is because since the block is already traveled down the ramp, the only energy it can have is kinetic.

01:19 So the final energy is 1 half mvf squared.

01:26 The initial energy, since the ramp that it travels down, is frictionless, the energy with which it goes into the rough patch of the ground is going to be equal to the kinetic energy that it has at the bottom of the ramp, which is in turn going to be equal to the total gravitational energy that has at the top of the ramp.

01:49 And that is equal to mass of the block times acceleration due to gravity times the height of the ramp, which is what we're trying to solve.

01:59 And this is all equal to the non -conservative work done by friction.

02:07 So now we just need to substitute in an expression for the non -conservative work done by friction.

02:15 And as usual, that is work done by friction is equal to the coefficient of kinetic friction times normal force.

02:26 And since we're on a flat surface when we have friction, the normal force is going to be simply equal to the force of gravity, assuming that the block's not flying into the air or sinking into the ground.

02:37 So normal force is m g.

02:42 And that's all times it's negative since it's opposing the motion of the object, and it's multiplied by the distance d that the object travels...

Question

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