A machine is supplied energy at a rate of 4,000 W and does useful work at a rate of 3,760 W. What is the efficiency of the machi
2 answers:
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(a) 196 N
The equation of the forces on the side of the cord where the force F is applied is:
(1)
where T is the tension in the cord.
On the other side of the cord, the equation of the forces on the canister is
(2)
where
m = 20 kg is the mass of the canister
is the acceleration of gravity
a is the acceleration
From (1),
Substituting into (2),
We want the canister to move at constant speed, so
a = 0
And therefore:
b) 2.0 cm
The cord is inextensible, this means that the acceleration of its parts are the same. Therefore, the acceleration of the free end must be the same as the acceleration of the canister: and this means that the two parts also cover the same distance in the same time.
Therefore, the free end of the cord must be moved exactly the same as the canister, by 2.0 cm.
c) 3.92 J, the same
The work done by the tension in the cord is
where
T is the tension
d = 2.0 cm = 0.02 m is the displacement
As we said in part (a), the tension in the cord is equal to the force applied to the free end:
T = F
So
T = 196 N
Therefore, the work done by the tension is
And since the force applied (F) is the same, then the work done by you when pulling the cord is exactly the same.
(d) -3.92 J
The weight of the canister is
However, the direction of the force of gravity is opposite to the displacement. Therefore, the work done by gravity is negative:
And substituting,
(e) Zero
The net work done on the canister can be simply calculated by adding the work done by the tension in the cord and the weight of the canister:
This is in agreement with the work-energy theorem, which states that the work done on an object is equal to its change in kinetic energy. In this situation, the canister is moving at constant speed, so its kinetic energy is not change: therefore,
(change in kinetic energy = 0)
and so, the work done on it is also zero.
(f) The pulley system changes the direction of the force applied
This is a simple pulley system, which means that the system does not multiply the force applied in input. In fact, the mechanical advantage of the system is
where:
is the output force, which is the weight of the canister
is the force in input, which is F
So, the mechanical advantage is 1:
From a point of view of energy, therefore, there is no advantage in this system.
However, the advantage offered by the pulley system concerns the direction of the force: in fact, it changes the direction of the applied force (which is F, downward) into the tension of the cord (which is upward on the canister).