The output of the nation's factories, mines and utilities expanded at its slowest monthly rate since January, the Federal Reserve Board reported yesterday.

The Fed said big increases in consumer durable goods like cars and appliances and a large rise in production of business equipment were partially offset by strikes in the coal and copper industries and by reduction in steel output.

The 0.5 per cent rise in overall industrial production in July followed a 0.7 per cent rise in June and a 0.9 per cent May advance.

Industrial production fell 0.8 per cent in January as cold weather and natural gas shortages closed factories and otherwise complicated production. But the economy rebounded in February and March.

Economists watch the rate of growth of industrial production carefully because increases in factory output usually translate quickly into a pickup in hiring.

Despite the severe early winter weather, the economy grew swiftly and unemployment was sharply reduced between November and July. Administration forecasters and most private economists think that the economy will grow more slowly for the rest of the year, with a consequent slowdown in improvement of the jobs picture.

Administration economists think that the 7 per cent advance in real gross national product recorded in the first half of the year will slow to about a 5 per cent annual rate in the last six months of the year. Industrial Production represents an important produced by American citizens after the value of all goods and services removing the impact of inflation.

While a slowing in the rate of growth of industrial production would seem to be consistent with a general slackening in overall economic growth, the Federal Reserve said yesterday that much of the slowdown was due to strikes.

The 0.5 per cent gain in industrial production works out to an annual rate of about 6.2 per cent. Over the last year industrial production rose 6.4 per cent. In the second quarter of the year industrial production was 2.8 per cent higher than in the first three months of 1977.

The Federal Reserve, the nation's central bank, said that output of consumer durable goods rose 1.6 per cent in July, "reflecting a large increase in automotive products and a more moderate rise in home goods."

Production of non-durable consumer goods - such as foods, tobacco products, apparel and paper - rose only 0.1 per cent. Overall, consumer goods production increased 0.6 per cent.

Output of business equipment, which had lagged earlier in the recovery, rose sharply in July as it has done in recent months. Output of business equipment was 1.2 per cent above June and 11.5 per cent above a year ago, the central bank reported.

Investment in plant and equipment must continue and even accelerate if the recovery is to continue at a fast-enough pace to get unemployment near 4 per cent by 1980 and put the federal budget back in balance by 1981 - two of the Carter administration's prime economic goals.

Output of intermediate products such as supplies for construction rose 0.8 per cent in July.

Production of durable materials "was reduced slightly in July, as further sizeable gains in the production of parts and components for consumer durable goods and business equipment were more than offset by weakness in basic metal materials, particularly steel."

Steel users - who faced a June 16 price increase on the industry's major product, flat-rolled products - may have ordered steel for their inventories in May and June and cut back orders in late June and July.

The major steel companies have announced price increases on tin mill products (used to make beverage containers) and structural shapes (used in heavy construction) to take effect Sept. 4. That may boost some steel production this month.