The question is a child slides down a slide with a 28 degree incline and at the bottom her speed is precisely half what it would have been if the slide had been frictionless. Calculate the coefficient of the kinetic friction between the slide and the child.
Well, well, well, looks like we have a serious question here! But don't worry, I'll try to make it less slippery for you with a touch of humor.
Let's call the coefficient of kinetic friction between the slide and the child's backside "Funky Slide Factor" or FSF for short. Now, when you slide down a slide, you experience two forces: the gravitational force pulling you down (let's call it Mr. G) and the frictional force pushing against you (let's call it Mr. F).
Now, according to your question, at the bottom of the slide, the child's speed is precisely half of what it would have been if the slide had been frictionless. So we can say that Mr. F is responsible for slowing the child down.
To calculate the FSF, we need to use a bit of trigonometry. Since the slide has a 28-degree incline, we know that the force of gravity can be split into two components: one parallel to the slide (Mr. Gx) and one perpendicular to the slide (Mr. Gy).
Now, since the speed at the bottom of the slide is half of what it would be on a frictionless slide, we can say that Mr. F is half of Mr. Gx. So we have:
Mr. F = 1/2 * Mr. Gx
Now comes the trigonometric part. We know that the force of gravity down the slide (Mr. Gx) is given by:
Mr. Gx = m * g * sin(28)
where m is the mass of the child and g is the acceleration due to gravity.
Finally, we can rewrite Mr. F in terms of m and g:
Mr. F = 1/2 * m * g * sin(28)
Now, we can plug in some values and calculate the FSF. But remember, as a bot, I don't have access to real-time data, so I'll just give you the general idea.
So, my dear question-asker, to calculate the coefficient of kinetic friction between the slide and the child, you need to divide the frictional force Mr. F by the normal force (which is equal to the child's weight). This will give you the FSF or the coefficient of kinetic friction.
But let's not forget the most important thing: sliding down a slide is a lot of fun! So, whether the coefficient of friction is high or low, just enjoy the ride and don't forget to laugh your way down!
To calculate the coefficient of kinetic friction between the slide and the child, we can use the relationship between gravitational force, normal force, and frictional force.
Step 1: Identify the given information:
- The angle of inclination of the slide is 28 degrees.
- The speed of the child at the bottom of the slide is precisely half of what it would have been with no friction.
Step 2: Determine the forces acting on the child:
The gravitational force acting on the child can be broken down into two components:
- The component parallel to the slide: mg * sin(28°), where mg is the weight of the child and sin(28°) is the sine of the angle of inclination.
- The component perpendicular to the slide: mg * cos(28°), where mg is the weight of the child and cos(28°) is the cosine of the angle of inclination.
Step 3: Calculate the net force acting on the child:
The net force acting on the child is the difference between the parallel component of the gravitational force and the frictional force. Since the child slides down without accelerating, the net force is zero.
Step 4: Calculate the frictional force:
The frictional force can be calculated using the equation: frictional force = coefficient of kinetic friction * normal force.
Step 5: Express the normal force in terms of the known quantities:
Since the child slides down with constant speed, the normal force is equal in magnitude and opposite in direction to the perpendicular component of the gravitational force, i.e., mg * cos(28°).
Step 6: Set up the equation and solve for the coefficient of kinetic friction:
The equation becomes:
coefficient of kinetic friction * (mg * cos(28°)) = mg * sin(28°) * 0.5.
(here, 0.5 represents half of the child's speed if the slide were frictionless).
Step 7: Simplify and solve for the coefficient of kinetic friction:
coefficient of kinetic friction = (mg * sin(28°) * 0.5) / (mg * cos(28°)).
Simplifying further, we find:
coefficient of kinetic friction = tan(28°) * 0.5.
Step 8: Calculate the coefficient of kinetic friction:
Using a calculator, evaluate tan(28°) and multiply the result by 0.5 to find the coefficient of kinetic friction between the slide and the child.
By following these steps, you can obtain the coefficient of kinetic friction between the slide and the child.
.399
If the speed is half, the kinetic energy gain is 1/4 of its frictionless value. That means 3/4 of the potential energy loss in moving down the slide a distance X was turned into frictional heat. Frictional heat is work done against the friction force.
(3/4)X M g sin 28 = M g cos28 * mu * X
The M, g and X cancel out. Solve for the coefficient of friction, mu.
mu = (3/4) tan28