In a machine a cylinder with a mass of 80.0 g is pushed down into a hole with a spring at the
bottom as shown in Figure 1. When the spring is uncompressed the cylinder rests on top of it with
its end just at the entrance to the hole. The cylinder is pushed by a rod which is angled at 30◦ as
shown. The rod exerts a force of 10.0 N parallel to the line that the rod points along. The walls of
the hole are well oiled so you can ignore friction. The spring has a stiffness of 1200 N/m. Getting
the timing of the machine right requires that we know the speed of the cylinder when the spring is
6.00 cm long as shown.
(a) Choose your system and define your axes clearly.
(b) Draw an energy bar chart for the situation.
(c) Solve for the speed of the cylinder when the spring is 6.00 cm long.
(d) Calculate the height of each of the “bars” in your energy bar chart. Is there one of the types
of energy that you used which is very small compared to the others? Recalculate the answer
to c) neglecting this contribution to see how good the approximation is. (In practice when
objects in machines are moving fast and there are spring forces involved the approximation
you have just found is a very good approximation.)
To solve this problem, we need to follow the given steps:
(a) Choosing your system and defining your axes clearly:
- Let's choose the system to be the cylinder and the spring.
- Define the x-axis to be parallel to the rod and pointing in the direction of the force exerted by the rod.
- Define the y-axis to be perpendicular to the x-axis, pointing upward.
(b) Drawing an energy bar chart for the situation:
- The energy bar chart will help us visualize and analyze the different types of energy involved in the system.
- The types of energy involved in this system are:
1. Gravitational potential energy (PEg)
2. Elastic potential energy (PEe)
3. Kinetic energy (KE)
(c) Solving for the speed of the cylinder when the spring is 6.00 cm long:
- To solve for the speed, we need to apply the principle of conservation of mechanical energy in this case.
- At the top before the cylinder starts compressing the spring, the only energy present is the gravitational potential energy (PEg).
- When the spring is compressed by 6.00 cm, the energy is converted from gravitational potential energy (PEg) to elastic potential energy (PEe) and kinetic energy (KE).
- At the bottom, when the spring is fully compressed, all the initial gravitational potential energy (PEg) is converted into the elastic potential energy (PEe) stored in the spring and the kinetic energy (KE) of the cylinder.
- At this point, no further work is done on the cylinder, so we can equate the initial gravitational potential energy (PEg) to the sum of the final elastic potential energy (PEe) and kinetic energy (KE).
- Using the energy equation, we can solve for the speed of the cylinder.
(d) Calculating the height of each of the "bars" in your energy bar chart:
- The height of each bar in the energy bar chart represents the magnitude of each energy type.
- Calculate the initial gravitational potential energy (PEg) based on the mass of the cylinder and the height it is initially lifted.
- Calculate the final elastic potential energy (PEe) based on the stiffness of the spring and the displacement of the spring when it's compressed to 6.00 cm.
- Calculate the kinetic energy (KE) based on the mass of the cylinder and the speed we found in part (c).
- By examining the heights of each bar, we can determine if any energy type is significantly smaller compared to the others. If so, we can neglect this contribution in the calculations.