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Total 0/2 A 3.5-kg block slides with a speed of 1.6 m/s on a frictionless, horizontal surface until it encounters a spring.
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To find the answer to this question, we need to use concepts of conservation of mechanical energy and Hooke's Law.
1. Start by calculating the initial kinetic energy of the block. The formula to find kinetic energy is KE = (1/2) * mass * velocity^2. Given that the mass of the block is 3.5 kg and its speed is 1.6 m/s, we can calculate the initial kinetic energy as KE_initial = (1/2) * 3.5 kg * (1.6 m/s)^2.
2. Since there is no friction, the initial kinetic energy will be conserved as potential energy stored in the spring when it is compressed. Therefore, we can equate the initial kinetic energy to the potential energy stored in the spring. The formula for potential energy stored in a spring is PE = (1/2) * k * x^2, where k is the spring constant and x is the displacement from equilibrium. For simplicity, we will assume that the block comes to rest when the spring is compressed and reaches maximum displacement, so x will be the maximum compression of the spring.
3. Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from equilibrium. The formula is F = -k * x, where F is the force exerted by the spring and k is the spring constant. Since the motion stops when the block is at maximum compression, the force exerted by the spring is balanced by the force of gravity acting on the block. Therefore, we can equate the force of gravity to the force exerted by the spring: m * g = k * x.
4. Rearrange the equation from step 3 to solve for x: x = (m * g) / k. Substituting the known values, the maximum displacement of the spring can be calculated as x = (3.5 kg * 9.8 m/s^2) / k.
5. Substitute the value of x from step 4 into the formula for potential energy (PE) to find the energy stored in the spring at maximum compression: PE = (1/2) * k * x^2.
By following these steps, you should be able to determine the maximum potential energy stored in the spring when the block comes to rest.