Math  /  Algebra

QuestionFigure it thomure 6
In Figure 1, a person uses a pulley to lift a bell. The person pulls down on the rope at a constant speed. Power P1P_{1} is delvered to the bell and it moves upward at a constant speed. In Figure 2, the person uses a double pulley. The person pulls down on the rope at the same constant speed. The bell again moves upward at a constant speed, but the speed of the bell is half the speed of the bell in Figure 1 . The power delivered to the bell in Figure 2 is P2P_{2}. Which of the following correctly compares P2P_{2} to P1P_{1} and provides a valid justification? (A) P2<P1P_{2}<P_{1}, because in Figure 2 the force exerted on the bell is the same but it is moving with less speed than in Figure 1. (B) P2<P1P_{2}<P_{1}, because in Figure 2 the force exerted on the bell is less than in Figure 1. (C) P2=P1P_{2}=P_{1}, because in Figure 2 the speed is half that in Figure 1 , but the force on the bell is twice that in Figure 1. (D) P2=P1P_{2}=P_{1}, because in Figure 2 the person is doing the same amount of work on the bell as in Figure 1.

Studdy Solution

STEP 1

1. The speed of the bell in Figure 2 is half the speed of the bell in Figure 1.
2. Power is defined as the rate at which work is done or energy is transferred.
3. The force exerted by the pulley system in Figure 2 may differ from that in Figure 1 due to mechanical advantage.

STEP 2

1. Define the relationship between power, force, and speed.
2. Analyze the mechanical advantage provided by the double pulley system in Figure 2.
3. Compare the power delivered to the bell in both scenarios.

STEP 3

Power P P is given by the equation P=Fv P = F \cdot v , where F F is the force exerted on the bell and v v is the velocity of the bell.

STEP 4

In Figure 2, the double pulley system provides a mechanical advantage, which means the force exerted by the person is distributed over a greater distance, effectively doubling the force exerted on the bell compared to Figure 1.

STEP 5

In Figure 2, the speed of the bell is half, but the force exerted on the bell is doubled due to the mechanical advantage. Therefore, the power P2=2Fv2=Fv=P1 P_2 = 2F \cdot \frac{v}{2} = F \cdot v = P_1 .
The correct comparison is:
(C) P2=P1 P_{2}=P_{1} , because in Figure 2 the speed is half that in Figure 1, but the force on the bell is twice that in Figure 1.

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