The largest grand piano in the world is really grand. Built in London, it has a mass of 1.25 × 103 kg. Suppose
a pianist finishes playing this piano and pushes herself from the piano so that she rolls backwards with a
speed of 1.4 m/s. Meanwhile, the piano rolls forward so that in 4.0 s it travels 24 cm at constant velocity.
Assuming the stool that the pianist is sitting on has a negligible mass, what is the pianist’s mass?
76
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To solve this problem, we can use the principle of conservation of momentum. The momentum before the pianist pushes herself from the piano is equal to the momentum after she pushes off.
Let's denote the mass of the pianist as "m" and the mass of the piano as "M". The velocity of the pianist before she pushes off is 0 since she is initially at rest. The velocity of the piano is unknown.
The momentum before the pianist pushes off is given by:
M * 0 + m * 0 = 0
The momentum after the pianist pushes off is given by:
M * v + m * (-v)
where "v" is the velocity of the piano after the push, and "-v" is the negative velocity of the pianist after the push (since she is moving in the opposite direction).
According to the problem, in 4.0 seconds, the piano travels 24 cm (or 0.24 m) at constant velocity. We can use this information to find the velocity of the piano.
Velocity (v) = Distance (d) / Time (t)
v = 0.24 m / 4.0 s
v = 0.06 m/s
So the momentum after the pianist pushes off is:
M * 0.06 + m * (-0.06) = 0
Since the total momentum before and after the push must be equal, we can set these equations equal to each other:
0 = 0.06M - 0.06m
Now, let's consider the information given about the mass of the piano:
Mass of the piano (M) = 1.25 × 10^3 kg
Substituting this value into the equation, we get:
0 = 0.06(1.25 × 10^3) - 0.06m
Simplifying the equation:
0 = 75 - 0.06m
Rearranging the equation to solve for "m":
0.06m = 75
m = 75 / 0.06
m ≈ 1250 kg
Therefore, the pianist's mass is approximately 1250 kg.
To find the mass of the pianist, we can use the principle of conservation of momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, assuming no external forces are acting on the system.
In this case, the initial momentum is zero because both the pianist and the piano are stationary. The final momentum is the sum of the momentum of the pianist and the momentum of the piano after the push.
Let's assign variables to the unknowns:
- Mass of the pianist: m (to be calculated)
- Velocity of the pianist: v (given as 1.4 m/s)
- Mass of the piano: Mp (given as 1.25 × 10^3 kg)
- Velocity of the piano: Vp (to be calculated)
- Change in position of the piano: Δx (given as 24 cm = 0.24 m)
- Time taken: t (given as 4.0 s)
The momentum before the push is zero, so the total momentum after the push must also be zero. Mathematically, we can write:
(m * v) + (Mp * Vp) = 0
We also know that the piano rolls forward at a constant velocity, which means its momentum remains constant. We can use this information to find the velocity of the piano, Vp.
The momentum of the piano is given by its mass multiplied by its velocity: Mp * Vp.
Since the piano is rolling at a constant velocity, we know its change in position, Δx, divided by the time taken, t, gives us its velocity:
Vp = Δx / t
Vp = 0.24 m / 4.0 s
Vp = 0.06 m/s
Substituting this value of Vp into the momentum equation:
(m * v) + (Mp * 0.06) = 0
(m * 1.4) + (1.25 × 10^3 * 0.06) = 0
1.4m + 75 = 0
1.4m = -75
m = -75 / 1.4
m ≈ -53.57 kg
Since mass can't be negative, we must disregard the negative sign. Therefore, the mass of the pianist is approximately 53.57 kg.