The following is the gist from chapters - 3, 4 & 5 dealing with IS-LM model from Macroeconomics, Blanchard & Johnson, 6th edition.
Equilibrium in the financial market implies an increase in income leads to an increase in the interest rate, therefore the LM curve is upward sloping. The LM curve is derived at a given level of (real) money supply. It is further assumed that price level is fixed in the short run. further, it is assumed that central bank changes money supply through open market operations (OMOs). When the central bank buys bonds, bond prices goes up & interest rate goes down. Conversely, when the central bank sells bonds, bond prices goes down and interest rate goes up. Hence interest rate is directly dependent on the OMOs.
What I could not understand is why the interest rate is changing because of change in income, since there is no change in money supply and hence no OMO is going on?
Also, why would someone will buy the bond from central bank when the bond prices are going down?
Edit: (related to fixed money supply, first question) Since there is no interest rate on check-able deposits and bank money apart from reserves are used to purchase bonds, not for lending, people to people borrowing is an option. However, it does not seem to be a plausible due to the following reasons. The Money demand shift to right for an increase in income and intersects the Money supply curve at a higher point leading to increase in interest rate. Increase in income will suppress the people to people borrowing, leading to opposite of the required result, lower interest.
(related to variable money supply, second question) consumers who have already invested may hold on their bonds but may not purchase new bonds with the extra income. So the interest rate should not change. Moreover, if consumers buy and sell bonds among themselves without Central Bank Intervention and make major changes in the interest rate, they may not cooperate. e.g. CB decides to decrease money supply but consumers may not give up thier bonds as their income may have increase recently.