- #1
User1265
- 29
- 1
- Homework Statement
- An object of mass m and height d=4cm is inside a uniform hollow cuboid box of mass M=200g that sits on a flat table. The middle of the top of the object is attached to a spring of stiffness k=19.62N/m and natural length L=30cm which hangs from the middle of the internal ceiling of the box. Intially, the object is held against the inside base of the box by a release mechanism so that the spring is stretched by length x=20cm.
When the release mechanism is activated, the object is released from the inside base of the box and it is pulled upwards by the spring. Given that the spring compression is just large enough to make the box leave the table (and jump very slightly into the air), calculate m in grams. You may take the acceleration due to gravity as 9.81m/s2.
- Relevant Equations
- F=kx
Conseravtion of Energy
Upon reading, realized that equations of consevation of energy would be of use for this question.
I considered the energy of the system - the spring mass and box - taking into account that there will be Elastic potential energy of the spring, and we can arbitrarily set the intial positon(A) of the mass m, at base of the box, as h=0, as we are concerned only with the change in gravitational potential energy.
E(A) = (1/2)kx^2
I understand there must be another critical moment just before the box lifts, which is when sufficient compression, 'y' of the spring by mass m. Call this position B,
Mass m will now posses, gravitational potential energy as well as elastic potential energy, Q - but why not Kinetic energy? Is the sufficient compression the maximum compression in order to the box to lift? I don't really understand why Kinetic energy wouldn't be included in equation for energy at position B :
E(B) = (1/2)ky^2 + mgh
where h = y+x
as moved through an distance x to get back to natural length from extension, and distance y past natural length to compression.
E(B) = (1/2)ky^2 + mg(y+x)
The potential energy of the box is zero for both moments A and B.
To consider the critical moment the box is about to lift off the table, the forces acting on the box need to be analysed.
At the instant the box lifts, the normal reaction force on the box becomes zero, and the upwards force exerted on the box due to the compression of the spring by mass m, must be equal to the weight of the box.
Q2) But what I don't understand why the upwards force is 'ky' - I know that elastic force of the srping act towards the natural length, to try to restore the springs natural length (its own equlibrium) , so surely if the mass is exerting a force upwards on the spring to compress it by 'y', the spring would exert an elastic force of ky in the downards direction to try to restore the natural length, not upwards.
I considered the energy of the system - the spring mass and box - taking into account that there will be Elastic potential energy of the spring, and we can arbitrarily set the intial positon(A) of the mass m, at base of the box, as h=0, as we are concerned only with the change in gravitational potential energy.
E(A) = (1/2)kx^2
I understand there must be another critical moment just before the box lifts, which is when sufficient compression, 'y' of the spring by mass m. Call this position B,
Mass m will now posses, gravitational potential energy as well as elastic potential energy, Q - but why not Kinetic energy? Is the sufficient compression the maximum compression in order to the box to lift? I don't really understand why Kinetic energy wouldn't be included in equation for energy at position B :
E(B) = (1/2)ky^2 + mgh
where h = y+x
as moved through an distance x to get back to natural length from extension, and distance y past natural length to compression.
E(B) = (1/2)ky^2 + mg(y+x)
The potential energy of the box is zero for both moments A and B.
To consider the critical moment the box is about to lift off the table, the forces acting on the box need to be analysed.
At the instant the box lifts, the normal reaction force on the box becomes zero, and the upwards force exerted on the box due to the compression of the spring by mass m, must be equal to the weight of the box.
Q2) But what I don't understand why the upwards force is 'ky' - I know that elastic force of the srping act towards the natural length, to try to restore the springs natural length (its own equlibrium) , so surely if the mass is exerting a force upwards on the spring to compress it by 'y', the spring would exert an elastic force of ky in the downards direction to try to restore the natural length, not upwards.
