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 solve this problem, 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, provided that no external forces act on the system.
Let's break down the given information:
1. Mass of the piano (m1) = 1.25 × 10^3 kg
2. Speed of the pianist (v2) = 1.4 m/s
3. Time taken by the piano to travel 24 cm (t) = 4.0 s
Let's assume the mass of the pianist is m2.
Since no external forces act on the system (piano + pianist), the total momentum before and after the event should be equal.
The momentum before the event (momentum of the piano) is given by:
P1 = m1 * v1
where v1 is the velocity of the piano, which can be calculated using the given distance traveled and time:
v1 = distance/time
Substituting the values, we get:
v1 = 24 cm / (4.0 s) = 24 cm/s
Now, let's calculate the momentum before the event (P1):
P1 = m1 * v1
Substituting the values, we get:
P1 = (1.25 × 10^3 kg) * (24 cm/s)
To perform the calculation, we need to convert the centimeters to meters:
P1 = (1.25 × 10^3 kg) * (0.24 m/s) = 300 kg·m/s
The momentum after the event (momentum of the pianist) is given by:
P2 = m2 * (-v2)
Note that we need to use a negative sign for the velocity since the pianist is moving in the opposite direction.
Substituting the values, we get:
P2 = m2 * (-1.4 m/s)
Since the total momentum before and after the event is conserved, we can write:
P1 = P2
Substituting the values, we get:
300 kg·m/s = m2 * (-1.4 m/s)
To find the mass of the pianist (m2), we can rearrange the equation:
m2 = 300 kg·m/s / (-1.4 m/s)
Calculating this, we get:
m2 ≈ -214.29 kg
Since mass cannot be negative, we can ignore the negative sign in this context. Therefore, the mass of the pianist is approximately 214.29 kg.
Please note that in a realistic scenario, the mass of a pianist would not be negative. However, in this mathematical calculation, the negative sign indicates the opposite direction of the pianist's velocity.