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?
To find the mass of the pianist, we can use the principle of conservation of momentum.
The momentum before the pianist pushes away from the piano is equal to the momentum after she pushes away. The momentum of an object is the product of its mass and velocity.
Let's denote the mass of the pianist as m and her initial velocity as v1. The mass of the piano is given as 1.25 × 10^3 kg, and the pianist rolls backward with a speed of 1.4 m/s. Since there are no external forces acting on the system, we can assume the momentum is conserved.
The momentum before pushing away: m * v1
The momentum after pushing away: -(1.25 × 10^3 kg + m) * v1/1.25 × 10^3 kg
Setting these two equal to each other and solving for m:
m * v1 = -(1.25 × 10^3 kg + m) * v1/1.25 × 10^3 kg
Simplifying the equation:
m = -(1.25 × 10^3 kg + m)
Distributing the negative sign:
m = -1.25 × 10^3 kg - m
Moving m to one side:
2m = -1.25 × 10^3 kg
Dividing by 2:
m = -1.25 × 10^3 kg/2
m = -625 kg
However, mass cannot be negative in this context. Therefore, we made an error in our calculations. Let's reassess the problem.
Given that the piano rolls forward in 4.0 seconds and travels 24 cm, we can calculate the velocity of the piano:
Velocity of the piano = Distance/Time = 24 cm/4.0 s = 0.24 m/4.0 s = 0.06 m/s
Since the piano moves forward, its velocity is positive.
Now, let's reconsider the conservation of momentum:
m * v1 = (1.25 × 10^3 kg + m) * 0.06 m/s
Simplifying the equation:
m * v1 = 1.25 × 10^3 kg * 0.06 m/s + m * 0.06 m/s
m * v1 = 75 kg + 0.06 m
Subtracting 0.06 m from both sides:
m * v1 - 0.06 m = 75 kg
Factoring out m on the left side:
m * (v1 - 0.06) = 75 kg
Dividing both sides by (v1 - 0.06):
m = 75 kg / (v1 - 0.06)
We need the value of v1 to calculate the mass. Unfortunately, the initial velocity of the pianist is not provided in the given information. Without knowing v1, we cannot determine the exact mass of the pianist.
To solve this problem, we need to use the principle of conservation of momentum. According to this principle, the total momentum before the pianist pushes herself from the piano should be equal to the total momentum after she pushes herself.
Let's denote:
- m₁ as the mass of the pianist,
- m₂ as the mass of the piano, and
- v₁ as the speed of the pianist after pushing herself.
The initial momentum (before the push) is given by the equation:
p₁ = m₁ * v₁
The final momentum (after the push) can be calculated using the velocity and mass of the piano. Since the piano rolls at a constant velocity (constant speed), its momentum remains constant throughout the 4.0 second time interval.
Therefore, the final momentum is given by:
p₂ = m₂ * v₂
Since the final momentum is equal to the initial momentum, we have:
m₁ * v₁ = m₂ * v₂
We can solve for m₁ by rearranging the equation:
m₁ = (m₂ * v₂) / v₁
Given:
m₂ = 1.25 × 10³ kg,
v₂ = (24 cm) / (4.0 s) = 6.0 cm/s = 0.06 m/s,
v₁ = -1.4 m/s (the pianist rolls backward with a negative velocity).
Substituting the values into the equation, we can calculate the mass of the pianist:
m₁ = (1.25 × 10³ kg * 0.06 m/s) / -1.4 m/s
m₁ = -3.75 × 10¹ kg
Since mass cannot be negative (it is a scalar quantity), the mass of the pianist is approximately 3.75 × 10¹ kg.