- #1

- 474

- 13

- Homework Statement
- A spring has a relaxed length of .2m and its spring stiffness is 8 N/m. You glue a .06 kg block to the top of the spring, and push the block down, compressing the spring until its total length is .1 m. You make sure the block is at rest and then quickly move your hand away. the block begins to move upward, because the upward force on the block by the spring is greater than the downward force on the block by the Earth. Make a graph of y vs time for the block during a .3 s interval after you release the block.

- Relevant Equations
- F{spring} = -K*S*L

F = ma

y = y_o+v_o*t+.5*a*t^2

p_f = p_i + F_net*t

Just to be clear, this isn't a homework problem. it is an example problem found on page 68 of the text "Matter and Interactions" 4th edition. The solution is given in the book, but I'm having difficulty following their reasoning.

according to the book the net force is not constant, therefore we can only get estimates by calculating iteratively.

This makes sense. gravity is constant -9.81 m/s2, but as the spring expands, the amount of force it will exert on the block becomes smaller until it reverses direction. and becomes larger until it overcomes the momentum of the block and then repeat until it overcomes gravity.

The problem is that the acceleration is varying.

So I thought, I'll assume over such a short period of time, a_avg = a_initial. let me find the net Force, use that to find the initial acceleration, then find the position after .1s. repeat.

The book solves this by assuming that over a short period of time, v_avg = v_final. so finding the net force, use the initial momentum and that to find final momentum. find the final velocity from that. Because they assume that v_final is the average velocity, they plug that into the kinematic equation y_i + v_final*t.

their final position after one iteration is y=.135 m. (.1 + .035)

My final position after one iteration is y = .118 m. (.1 + 1/2*.035)

my question is: is there an issue with how I chose to solve the problem? Am I wrong? or is my estimate a better estimate since I assumed the acceleration was constant over the iteration and factored in changing velocity, and they assumed the velocity was constant over the iteration.

I've had this issue again and again with the examples in this book. they love to assume that the final velocity is the average velocity in their estimates even though we know the acceleration. to me it seems like it makes more since to estimate the average acceleration in a problem like this than the velocity.

according to the book the net force is not constant, therefore we can only get estimates by calculating iteratively.

This makes sense. gravity is constant -9.81 m/s2, but as the spring expands, the amount of force it will exert on the block becomes smaller until it reverses direction. and becomes larger until it overcomes the momentum of the block and then repeat until it overcomes gravity.

The problem is that the acceleration is varying.

So I thought, I'll assume over such a short period of time, a_avg = a_initial. let me find the net Force, use that to find the initial acceleration, then find the position after .1s. repeat.

The book solves this by assuming that over a short period of time, v_avg = v_final. so finding the net force, use the initial momentum and that to find final momentum. find the final velocity from that. Because they assume that v_final is the average velocity, they plug that into the kinematic equation y_i + v_final*t.

their final position after one iteration is y=.135 m. (.1 + .035)

My final position after one iteration is y = .118 m. (.1 + 1/2*.035)

my question is: is there an issue with how I chose to solve the problem? Am I wrong? or is my estimate a better estimate since I assumed the acceleration was constant over the iteration and factored in changing velocity, and they assumed the velocity was constant over the iteration.

I've had this issue again and again with the examples in this book. they love to assume that the final velocity is the average velocity in their estimates even though we know the acceleration. to me it seems like it makes more since to estimate the average acceleration in a problem like this than the velocity.