A workshop compressor is driven by a belt connecting a motor pulley of radius 5 cm to a compressor pulley of radius 20 cm, shown in the diagram. The motor t...
Question 1 Report
A workshop compressor is driven by a belt connecting a motor pulley of radius 5 cm to a compressor pulley of radius 20 cm, shown in the diagram. The motor turns at 1200 revolutions per minute (rpm), and a pulley's radius multiplied by its rpm stays constant along the belt.
- Write the ratio of the two radii in its simplest form. (1)
- Find the speed of the compressor pulley, in rpm. (2)
- The compressor pulley needs to complete 900 revolutions. Find how long this takes, in seconds. (2)
- State the new compressor pulley radius needed to double its speed, keeping the motor speed fixed, and justify your answer. (1)
Answer Details
This question tests inverse proportion between the radius and the speed of two pulleys joined by a belt: a smaller pulley must spin faster to keep the belt moving at the same speed, so radius \(\times\) rpm stays constant.
(a) The two radii are 5 cm and 20 cm, giving the ratio
\[ 5 : 20 = 1 : 4 \] [1 mark]
(b) Since radius \(\times\) rpm is the same for both pulleys on the belt,
\[ 5 \times 1200 = 6000 \]
so the compressor pulley's speed is
\[ 6000 \div 20 = 300 \text{ rpm} \] [2 marks]
(c) At 300 rpm, the time to complete 900 revolutions is
\[ 900 \div 300 = 3 \text{ minutes} = 180 \text{ seconds} \] [2 marks]
(d) Doubling the compressor's speed to 600 rpm, with the same constant of 6000, needs a radius of
\[ 6000 \div 600 = 10 \text{ cm} \]
This is exactly half the original 20 cm radius, which fits the inverse relationship: halving the radius doubles the rpm, since radius and speed multiply to give a fixed number for a given motor speed. [1 mark]
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