James,
You can control how the computer responds to power requirements and a failing charge (battery is low).
The following is for Windows 8 -- Windows 7 is probably quite similar. References follow.
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Windows key + X > P (opens the Control Panel) > Power Options > Change advanced power settings >
In "Power Settings" / Advanced settings >>
Select in the Drop Down Menu box your "Active" Power option >
The menu responds by bringing up the settings for the Active Power Plan >
(wording varies, example: Balanced (Active)): >>
Scroll down in the menu and look for "Battery" >
Click on the plus sign next to "Battery" >
The menu reponds by opening the Battery settings sub-menu >>
Click on each setting, (for example, Critical battery action) >
Select the preferred action for "Critical battery" >
Under "Low battery level" > Select the percentage at which the selected action will be performed.
Continue as desired, setting each battery setting.
Click OK when finished to "set" the new actions.
Reference:
Managing Power Options (Windows 8)
Setting Power Button Behavior (Windows 7)
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It is possible that the battery is to blame, it may not be charging properly in spite of your testing results.
It is possible the fault lies with the AC charger. Chargers usually work or NOT, but they can also show signs of stress if the charger is under powered (lower) wattage than your system needs.
Using and Testing the AC Power Adapter
Purchasing a Replacement AC Adapter
As to your question whether replacing the battery will solve your issue: I do not know the answer to that. I do not think one hour to one hour thirty minutes on a charge result is acceptable. If the situation were mine to solve, I would investigate replacing both the battery and the charger.
I hope you solve the issue and that the information I have provided helps you with the decision process.
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