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.

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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 entirely into kinetic energy at point B. Use the conservation of mechanical energy: \( m g h = \frac{1}{2} m v_B^2 \), where \( h \) is the vertical height of the quarter-circle (equal to the radius, 2.0 m), \( m \) is the mass of the block (1.0 kg), and \( v_B \) is the velocity at point B.

Step 2: Solve for \( v_B \) in the equation \( m g h = \frac{1}{2} m v_B^2 \). Cancel out \( m \) on both sides, and rearrange to get \( v_B = \sqrt{2 g h} \). Substitute \( g = 9.8 \ \text{m/s}^2 \) and \( h = 2.0 \ \text{m} \) to find \( v_B \).

Step 3: Verify the transition from point B to point C. The block moves horizontally over a distance of 3.0 m with a coefficient of kinetic friction \( \mu_k = 0.25 \). The work done by friction is \( W_f = -f_k d \), where \( f_k = \mu_k m g \) is the kinetic friction force and \( d = 3.0 \ \text{m} \) is the distance. This work reduces the block's kinetic energy.

Step 4: Write the energy equation for the motion from B to C: \( \frac{1}{2} m v_B^2 + W_f = \frac{1}{2} m v_C^2 \), where \( v_C \) is the velocity at point C. Substitute \( W_f = -\mu_k m g d \) and solve for \( v_C \).

Step 5: Analyze the motion from point C to point D. The block compresses the spring by 0.20 m. Use the conservation of energy: \( \frac{1}{2} m v_C^2 = \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x = 0.20 \ \text{m} \) is the compression. Rearrange to solve for \( k \) if needed, or use this equation to verify the energy balance.

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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 to point B. Understanding this concept is crucial for calculating the block's velocity at point B, as it allows us to equate the potential energy lost to the kinetic energy gained.

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 concept is essential for analyzing the motion of the block as it travels from point B to point C, where friction will do work against the block's kinetic energy, affecting its speed and energy conservation.

Spring Potential Energy

Spring potential energy is the energy stored in a spring when it is compressed or stretched from its equilibrium position. The amount of energy stored in a spring is given by the formula U = (1/2)kx², where k is the spring constant and x is the compression or extension. This concept is vital for determining how much energy is transferred to the spring when the block compresses it by 0.20 m, impacting the overall energy balance in the system.

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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

A skier of mass m starts from rest at the top of a solid sphere of radius r and slides down its frictionless surface. If friction is present, does the skier fly off at a greater or lesser angle?

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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