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Chapter 10: Problem 56

. Playground Slide A girl whose weight is \(267 \mathrm{~N}\) slides down a \(6.1 \mathrm{~m}\) playground slide that makes an angle of \(20^{\circ}\) with the horizontal. The coefficient of kinetic friction between slide and child is \(0.10\). (a) How much energy is transferred to thermal energy? (b) If the girl starts at the top with a speed of \(0.457 \mathrm{~m} / \mathrm{s}\), what is her speed at the bottom?

Short Answer

Expert verified
(a) The energy transferred to thermal energy is 15.29 J. (b) The speed at the bottom is 5.38 m/s.

Step by step solution

01

- Calculate the Normal Force

The normal force ( R) can be calculated by considering the component of the girl's weight perpendicular to the slide. The component of the weight is R = W cos( 胃 ), where W is the weight (267 N) and 胃 is the angle (20掳). So the normal force R = 267 cos(20掳).
02

- Calculate the Frictional Force

The frictional force ( f_k ) is given by the coefficient of kinetic friction ( 渭_k ) multiplied by the normal force. Therefore, f_k = 渭_k R = 0.10 R.
03

- Calculate the Thermal Energy Transferred

The thermal energy ( E_{therm} ) transferred is equal to the work done by friction, which can be calculated as E_{therm} = f_k d , where d is the distance (6.1 m) the girl slides.
04

- Calculate the Component of Weight Parallel to the Slide

The component of the weight acting parallel to the slide is W_parallel = W sin( 胃 )= 267 sin(20掳).
05

- Calculate the Net Force Along the Slide

The net force ( F_{net} ) is the difference between the component of the weight parallel to the slide and the frictional force, F_{net} = W_parallel - f_k.
06

- Use Energy Conservation to Find Speed at the Bottom

The total mechanical energy at the top is the sum of kinetic and potential energy. At the bottom, the potential energy is zero, so using the work-energy principle: 1/2mv^2 = 1/2mv_0^2+mgh-E_{therm}. Solve for the final velocity v.
07

- Plug in Known Values

Substitute known values into the energy conservation equation, remembering to convert the height h (6.1m sin(20掳)), and solve for v.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Force
The normal force is a crucial concept to understand when discussing objects in motion on an inclined plane. It acts perpendicular to the surface and counters the object's weight. For the girl on the slide, her weight is 267 N, and the slide is at a 20掳 angle to the horizontal. This means we need to calculate the component of her weight that's perpendicular to the slide. Using the formula:
\( R = W \cos(\theta) = 267 \cos(20^{\circ}) \)
This calculation gives us the normal force acting on the girl, which influences other forces like friction.
Kinetic Friction
Kinetic friction acts against the motion of two surfaces sliding past each other. The amount of kinetic friction depends on the normal force and the coefficient of kinetic friction, which is a measure of how 'sticky' two surfaces are. Here, the coefficient of kinetic friction between the slide and the girl is 0.10. Using the normal force calculated earlier:
\( f_k = \mu_k R = 0.10 \cdot R \)
The frictional force will be essential in determining how much thermal energy is generated as she slides down.
Thermal Energy
As the girl moves down the slide, kinetic friction converts some of the mechanical energy into thermal energy. This is the energy that heats up the slide and the girl's clothes due to friction. The amount of thermal energy transferred is equal to the work done by the frictional force over the distance she slides:
\( E_{therm} = f_k \cdot d \)
By substituting the values of the frictional force and the distance (6.1 m), we can calculate the specific amount of thermal energy generated.
Energy Conservation
The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the girl starts with both potential and kinetic energy at the top of the slide. At the bottom, her potential energy has been converted into kinetic energy and thermal energy:
\( \frac{1}{2} mv^2 = \frac{1}{2} mv_0^2 + mgh - E_{therm} \)
This equation helps us determine her speed at the bottom by accounting for the initial energy and subtracting the energy lost to friction.
Mechanical Energy
Mechanical energy is the sum of potential and kinetic energy in an object. For the girl on the slide:
  • Potential energy depends on her height above the ground and is calculated as \( mgh \)
  • Kinetic energy depends on her speed and is calculated as \( \frac{1}{2} mv^2 \)
The total mechanical energy at any point is the sum of these two forms of energy. As she slides down, potential energy decreases while kinetic energy increases, offset by the work done against friction which is converted into thermal energy. This transfer and transformation of energy allow us to calculate her final speed at the bottom.

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Most popular questions from this chapter

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Ball and Spring Gun a ball of mass \(m\) is shot with speed \(v_{1}\) into the barrel of a spring gun of mass \(M\) initially at rest on a frictionless surface. The ball sticks in the barrel at the point of maximum compression of the spring. Assume that the increase in ther- mal energy due to friction between the ball and the barrel is negligible. (a) What is the speed of the spring gun after the ball stops in the barrel? (b) What fraction of the initial kinetic energy of the ball is stored in the spring?

Diatomic Molecule The potential energy of a diatomic molecule (a two-atom system like \(\mathrm{H}_{2}\) or \(\mathrm{O}_{2}\) ) is given by $$ U=\frac{A}{r^{12}}-\frac{B}{r^{6}} $$ where \(r\) is the separation of the two atoms of the molecule and \(A\) and \(B\) are positive constants. This potential energy is associated with the force that binds the two atoms together. (a) Find the equilibrium separation-that is, the distance between the atoms at which the force on each atom is zero. Is the force repulsive (the atoms are pushed apart) or attractive (they are pulled together) if their separation is (b) smaller and (c) larger than the equilibrium separation?

Potential Energy Function A single conservative force \(F(x)\) acts on a \(1.0 \mathrm{~kg}\) particle that moves along an \(x\) axis. The potential energy \(U(x)\) associated with \(F(x)\) is given by $$ U(x)=(-4.00 \mathrm{~J} / \mathrm{m}) e^{(-x /(4.00 \mathrm{~m}))} $$ At \(x=5.0 \mathrm{~m}\) the particle has a kinetic energy of \(2.0 \mathrm{~J}\). (a) What is the mechanical energy of the system? (b) Make a plot of \(U(x)\) as a function of \(x\) for \(0 \leq x \leq 10 \mathrm{~m}\), and on the same graph draw the line that represents the mechanical energy of the system. Use part (b) to determine (c) the least value of \(x\) and (d) the greatest value of \(x\) between which the particle can move. Use part (b) to determine (e) the maximum kinetic energy of the particle and (f) the value of \(x\) at which it occurs. (g) Determine the equation for \(F(x)\) as a function of \(x\). (h) For what (finite) value of \(x\) does \(F(x)=0 ?\)
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