Consider the track shown in Fig. 8–39. The section AB is one quadrant of a circle of radius 2.0 m and is frictionless. B to C is a horizontal span 3.0 m long with a coefficient of kinetic friction μₖ = 0.25. The section CD under the spring is frictionless. A block of mass 1.0 kg is released from rest at A. After sliding on the track, it compresses the spring by 0.20 m. Determine the stiffness constant k for the spring.

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Identify the energy transformations in the system: The block starts with gravitational potential energy at point A, which is converted into kinetic energy as it slides down the frictionless curve AB. From B to C, some of the kinetic energy is dissipated as thermal energy due to friction. Finally, the remaining kinetic energy is used to compress the spring at point D.

Write the expression for the total mechanical energy at point A: The gravitational potential energy at A is given by \( U_g = m g h \), where \( m = 1.0 \, \text{kg} \), \( g = 9.8 \, \text{m/s}^2 \), and \( h \) is the vertical height of the circular arc (calculated as the radius \( R = 2.0 \, \text{m} \)).

Determine the work done by friction from B to C: The work done by friction is \( W_f = -f_k d \), where \( f_k = \mu_k m g \) is the kinetic friction force, \( \mu_k = 0.25 \) is the coefficient of kinetic friction, and \( d = 3.0 \, \text{m} \) is the length of the horizontal span.

Write the energy conservation equation: The initial gravitational potential energy at A is equal to the sum of the work done by friction, the kinetic energy at C, and the elastic potential energy stored in the spring at D. The elastic potential energy is given by \( U_s = \frac{1}{2} k x^2 \), where \( x = 0.20 \, \text{m} \) is the compression of the spring and \( k \) is the spring constant to be determined.

Solve for the spring constant \( k \): Rearrange the energy conservation equation to isolate \( k \). Substitute the known values for mass, gravitational acceleration, height, friction, and spring compression to calculate \( k \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. In this scenario, the gravitational potential energy of the block at point A is converted into kinetic energy as it slides down the frictionless track and then into elastic potential energy when it compresses the spring. Understanding this concept is crucial for calculating the spring constant, as it allows us to relate the energies involved.

Kinetic Friction

Kinetic friction is the force that opposes the motion of two surfaces sliding past each other. It is quantified by the coefficient of kinetic friction (μₖ), which in this case is 0.25. This force acts on the block as it moves from point B to C, affecting its speed and energy. Knowing how to calculate the work done against friction is essential for determining the energy lost during this segment of the motion.

Hooke's Law

Hooke's Law describes the relationship between the force exerted on a spring and the displacement of the spring from its equilibrium position. It states that the force (F) is proportional to the displacement (x), expressed as F = -kx, where k is the spring constant. This law is fundamental for calculating the stiffness constant of the spring in the problem, as it relates the compression of the spring to the force exerted by it.

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Textbook Question

A spring ( k = 75 N/m) has an equilibrium length of 1.00 m. The spring is compressed to a length of 0.50 m and a mass of 2.0 kg is placed at its free end on a frictionless slope which makes an angle of 41° with respect to the horizontal (Fig. 8–41). The spring is then released. Now the incline has a coefficient of kinetic friction μₖ. If the block, attached to the spring, is observed to stop just as it reaches the spring’s equilibrium position, what is the coefficient of friction μₖ?

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Textbook Question

Consider the track shown in Fig. 8–39. The section AB is one quadrant of a circle of radius 2.0 m and is frictionless. B to C is a horizontal span 3.0 m long with a coefficient of kinetic friction μₖ = 0.25. The section CD under the spring is frictionless. A block of mass 1.0 kg is released from rest at A. After sliding on the track, it compresses the spring by 0.20 m. Determine the velocity of the block at point C.

Textbook Question

Early test flights for the space shuttle used a “glider” (mass of 980 kg including pilot). After a horizontal launch at 480 km/h at a height of 3200 m, the glider eventually landed at sea level with a speed of 210 km/h. What would its landing speed have been in the absence of air resistance?

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Textbook Question

A spring ( k = 75 N/m) has an equilibrium length of 1.00 m. The spring is compressed to a length of 0.50 m and a mass of 2.0 kg is placed at its free end on a frictionless slope which makes an angle of 41° with respect to the horizontal (Fig. 8–41). The spring is then released. If the mass is attached to the spring, how far up the slope will the mass move before coming to rest?

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Textbook Question