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In a truck-loading station at a post office, a small 0.200 -kg package is released from rest at point \(A\) on a track that is one-quarter of a circle with radius 1.60 \(\mathrm{m}\) (Fig. P7.65). The size of the package is much less than \(1.60 \mathrm{m},\) so the package can be treated as a particle. It slides down the track and reaches point \(B\) with a speed of 4.80 \(\mathrm{m} / \mathrm{s}\) . From point \(B\) , it slides on a level surface a distance of 3.00 \(\mathrm{m}\) to point \(C,\) where it comes to rest. (a) What is the coefficient of kinetic friction on the horizontal surface? (b) How much work is done on the package by friction as it slides down the circular arc from \(A\) to \(B\) ?

Step by step solution

01

Analyze the energy transformation from A to B

At point A, the package has gravitational potential energy and no kinetic energy. At point B, it has kinetic energy and possibly some lost energy due to friction. Using energy conservation between point A and B: \[ \text{Potential Energy at A} = \text{Kinetic Energy at B} + \text{Work Done by Friction} \]This translates to:\[ mgh = \frac{1}{2}mv^2 + W_{friction_{AB}} \]Where:- \( m = 0.200 \) kg (mass of the package)- \( g = 9.8 \) m/s² (acceleration due to gravity)- \( h = r = 1.60 \) m (height of the quarter-circle)- \( v = 4.80 \) m/s (speed at B)

02

Calculate the potential energy at A

The potential energy at the top of the track, point A, can be calculated using:\[ PE_A = mgh \]Substituting the known values:\[ PE_A = 0.200 \times 9.8 \times 1.60 \]

03

Calculate the kinetic energy at B

The kinetic energy at point B can be calculated using:\[ KE_B = \frac{1}{2}mv^2 \]Substituting the known values:\[ KE_B = \frac{1}{2} \times 0.200 \times (4.80)^2 \]

04

Calculate work done by friction from A to B

Using the calculated energies from Steps 2 and 3, the work done by friction (\(W_{friction_{AB}}\)) can be found by rearranging the energy equation:\[ W_{friction_{AB}} = PE_A - KE_B \]

05

Analyze motion from B to C

From B to C, the package slides on a horizontal surface and comes to rest. The work done by friction brings it to rest:\[ W_{friction_{BC}} = \text{friction force} \times \text{distance} = F_{friction} \times d \]The force of friction can be written as:\[ F_{friction} = \mu_k F_n \]Where \( \mu_k \) is the coefficient of kinetic friction, and the normal force \( F_n \) equals \( mg \) on a horizontal surface.

06

Solve for the coefficient of kinetic friction \(\mu_k\)

From B to C, the initial kinetic energy is converted completely into work done by the friction:\[ KE_B = W_{friction_{BC}} = \mu_k mgd \]Using the expression for work and friction we have:\[ \mu_k = \frac{KE_B}{mgd} \]

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Key Concepts

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

Kinetic Friction

Kinetic friction is the force that opposes the relative motion between two surfaces in contact. When the small package moves along the horizontal surface from point B to point C, kinetic friction acts against its motion and eventually brings it to a stop.

• The package's movement is initially maintained from the kinetic energy it has after leaving the circular track at point B, but as it slides, kinetic friction opposes this, converting kinetic energy into thermal energy.
• The magnitude of kinetic friction is proportional to the normal force (which is the weight of the package on a level surface here) and the coefficient of kinetic friction, denoted as \( \mu_k \).

Thus, determining the coefficient of kinetic friction is crucial in calculating how quickly the package comes to rest as it travels to point C.

Energy Conservation

Energy conservation is a principle stating that energy in a closed system remains constant, although energy can transform from one form to another.

• In this exercise, the package starts with gravitational potential energy at point A and converts some of it into kinetic energy at point B.
• The energy isn't completely transformed into kinetic energy due to kinetic friction, which does work on the package.

By understanding energy conservation, we can track how energy shifts within the system, first becoming kinetic energy as the package moves downhill and then being dissipated as thermal energy due to friction.

Gravitational Potential Energy

Gravitational potential energy is the energy possessed by an object due to its position in a gravitational field. At point A, while resting at the height equivalent to the radius of the circular track, the package holds gravitational potential energy.

• This potential energy is determined by the formula \( PE = mgh \), where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height above the reference point.
• As the package descends the quarter-circular track, this energy transforms primarily into kinetic energy at point B.

Gravitational potential energy sets the stage for understanding how high the package starts and how this elevation contributes to the kinetic energy it acquires while descending.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. At point B, the package's kinetic energy is at its maximum right before it travels along the horizontal surface.

• Kinetic energy is expressed as \( KE = \frac{1}{2}mv^2 \), where \( m \) is mass and \( v \) is velocity. The formula helps us calculate the energy the package must have gained while sliding down from point A.
• Once the package is on the horizontal plane, its kinetic energy is slowly converted into other forms of energy due to the work done by friction until it completely stops at point C.

Understanding kinetic energy is essential to identifying the speed and impact of moving objects like the package as they transition over different surfaces.