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.

Verified step by step guidance

Step 1: Analyze the motion from point A to point B. Since the section AB is frictionless and forms a quarter-circle, the block's potential energy at point A is converted into kinetic energy at point B. Use the conservation of mechanical energy: \( mgh = \frac{1}{2}mv^2 \), where \( h \) is the vertical height of the quarter-circle (equal to the radius, 2.0 m). Solve for the velocity \( v_B \) at point B.

Step 2: Analyze the motion from point B to point C. In this section, the block experiences kinetic friction. The work done by friction is \( W_{friction} = f_k \cdot d = \mu_k \cdot m \cdot g \cdot d \), where \( \mu_k = 0.25 \), \( m = 1.0 \) kg, \( g = 9.8 \, \text{m/s}^2 \), and \( d = 3.0 \) m. Subtract the work done by friction from the kinetic energy at point B to find the kinetic energy at point C.

Step 3: Use the kinetic energy at point C to calculate the velocity of the block at point C. The kinetic energy is given by \( KE = \frac{1}{2}mv^2 \). Rearrange this equation to solve for \( v_C \), the velocity at point C.

Step 4: Verify the energy conservation principle. Ensure that the total energy (potential energy at A, work done by friction, and spring compression energy) is consistent throughout the system. This step helps confirm the accuracy of the calculations.

Step 5: Summarize the relationships and equations used. The velocity at point C is determined by considering the energy transformations from point A to point C, accounting for the work done by friction in the horizontal section. This ensures a clear understanding of the problem-solving process.

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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 over time. In this scenario, the gravitational potential energy of the block at point A is converted into kinetic energy as it descends and moves along the track. Understanding this concept is crucial for calculating the block's velocity at point C, as it allows us to equate the initial potential energy with the kinetic energy at different points along the track.

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 point C, affecting its velocity. Knowing how to calculate the work done against friction is essential for determining the block's speed at point C.

Spring Potential Energy

Spring potential energy is the energy stored in a spring when it is compressed or stretched from its equilibrium position. It is given by the formula PE_spring = (1/2)kx², where k is the spring constant and x is the displacement from the equilibrium position. In this problem, understanding how the block compresses the spring by 0.20 m allows us to calculate the energy transferred to the spring, which is vital for analyzing the block's motion and velocity at point C.

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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 B.

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

You slide down an 8.0-m-high icy hill (≈ frictionless). At the bottom is a level stretch where the coefficient of kinetic friction is 0.30. How far would you travel across the level stretch?

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Textbook Question