Question

A 350-kg wooden raft floats on a lake. When a 77kg man stands on the raft, it sinks 3.9 cm deeper into the water. When he steps off, the raft oscillates for a while.

What is the frequency of oscillation? Express your answer using two significant figures.
What is the total energy of oscillation (ignoring damping)? Express your answer using two significant figures.

Answer 1

Frequency = 1.0 Hz

Energy = 59 J

Step-by-step explanation:

Steps

⇒ The raft alone weighs 350 kg and floats on the lake.

⇒ When a 77 kg man stands on the raft, it sinks 3.9 cm deeper.

⇒ When he steps off, the raft oscillates. We need to calculate the frequency and energy of oscillation.

Frequency Calculation:

When the man steps off, the raft experiences a restoring force due to the buoyant force. The raft will oscillate like a mass on a spring.

We can use Hooke's law to calculate the spring constant, k:

⇒ k = F/x = (F2 - F1)/(x2 - x1)

• F1 is the buoyant force on the raft alone
• F2 is the buoyant force when the man is on
• x1 is the original depth
• x2 is the new depth when the man is on

Plugging in the values:

⇒ k = (350kg9.8 - (350+77)kg9.8)/(0 - 0.039m)

⇒ = 293000 N/m

The frequency of oscillation is:

⇒ f = (1/2π)√(k/m)

Where m is the total mass (raft + man)

⇒ = 350 + 77 = 427 kg

⇒ f = (1/2π)√(293000/427) = 1.0

Energy Calculation:

The total energy of oscillation is:

⇒ E = (1/2)kx^2

Using x = original displacement = 0.039 m

And k = 293000 N/m determined above

⇒ E = (1/2)(293000)(0.039)^2

⇒ = 59 J

Answer 2

The frequency of oscillation is 7.4 Hz.

The total energy of oscillation is 515 J.

Step-by-step explanation:

Sure, I can help you with that.

**The frequency of oscillation**

The frequency of oscillation of the raft is given by the following formula:

```

f = 1 / (2 * pi) * sqrt(k / m)

```

where:

* `f` is the frequency of oscillation, in hertz (Hz)

* `k` is the spring constant, in newtons per meter (N/m)

* `m` is the mass of the raft, in kilograms (kg)

In this case, the spring constant is equal to the weight of the man divided by the distance that the raft sinks:

```

k = mg / x = (77 kg * 9.8 m/s^2) / 0.039 m = 21,000 N/m

```

The mass of the raft is 350 kg.

Plugging these values into the formula for the frequency of oscillation, we get:

```

f = 1 / (2 * pi) * sqrt(21,000 N/m / 350 kg) = 7.4 Hz

```

Therefore, the frequency of oscillation of the raft is **7.4 Hz**.

**The total energy of oscillation**

The total energy of oscillation of the raft is given by the following formula:

```

E = 1 / 2 * k * x^2

```

where:

* `E` is the total energy of oscillation, in joules (J)

* `k` is the spring constant, in newtons per meter (N/m)

* `x` is the amplitude of oscillation, in meters (m)

In this case, the amplitude of oscillation is equal to the distance that the raft sinks when the man steps off:

```

x = 0.039 m

```

Plugging these values into the formula for the total energy of oscillation, we get:

```

E = 1 / 2 * 21,000 N/m * (0.039 m)^2 = 5,100 J

```

Therefore, the total energy of oscillation of the raft is **5,100 J**.

**Answers**

The frequency of oscillation of the raft is **7.4 Hz**.

The total energy of oscillation of the raft is **5,100 J**.

bardAI