68) This question asks us how we can find the mass of the block. This means we need to come up with an equation that has the mass unknown. There are three equations that are possible to use.
Let's take a look at each of these! Each answer choice gives us 1 of the unknown variables and we need to solve for mass, so we need to find the equation with an unknown mass and 1 other unknown.
1. The first one uses the conservation of energy, which allows us to know that the potential energy at point Y will have the same magnitude as the kinetic energy at point O. We can further expand this formula.
The problem tells us that we know the value of k. Thus, in the orange equation, we don't know displacement(x) or speed(v), and we need to solve for mass(m). This equation has 3 unknown variables, so we cannot solve for mass.
2. This equation is a combination of Hooke's Law and Newton's 2nd Law. We are equating them because the Force of a spring should be equal to the product of its mass and acceleration. This equation has 3 unknowns: displacement(x), acceleration(a), and mass(m). Even if we are given either displacement(x) or acceleration(a), we wouldn't be able to solve for mass(m) unless we were given both.
3. Finally, we can look at this equation, which relates the period(T) of the spring's motion, to the mass(m) attached to the spring and the spring constant(k). The problem states that we know the value of k, so there are only 2 unknowns: period and mass. This means if we are given the period(T) of the spring's oscillation, we can solve for mass(m). However, none of the answer choices give us the period(T) of the oscillation. But when we take a closer look at answer choice E, we can tell that if we have the time it takes for the block to travel from X to Y, we can double that to find the period(T). This means this equation works and choice E is correct.
Answer: E
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