a steel strip camped at one end vibrate with a frequency of 20HZ and an applitude of 5mm, at the free end where a small mass of 2g is positioned,find the velocity of the end when passing through the Zoro position. the acceleration at maximum displacement. the maximum kinetic energy(Ek) of mass

Question

a steel strip camped at one end vibrate with a frequency of 20HZ and an applitude of 5mm, at the free end where a small mass of 2g is positioned,find the velocity of the end when passing through the Zoro position. the acceleration at maximum displacement. the maximum kinetic energy(Ek) of mass

Answer 1

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Answer 2

To find the velocity of the end when passing through the zero position, we can use the formula for the velocity of a simple harmonic motion:

v = ω * A

Where:
- v is the velocity
- ω is the angular frequency
- A is the amplitude

In this case, the frequency (f) is given as 20 Hz. We can find the angular frequency (ω) using the formula:

ω = 2π * f

Plugging in the values, we have:

ω = 2π * 20 = 40π rad/s

The amplitude (A) is given as 5 mm, which is equivalent to 0.005 m.

Now we can calculate the velocity:

v = 40π * 0.005
v ≈ 0.628 m/s

Therefore, the velocity of the end when passing through the zero position is approximately 0.628 m/s.

To find the acceleration at maximum displacement, we can use the formula:

a = ω^2 * x

Where:
- a is the acceleration
- ω is the angular frequency
- x is the displacement

In this case, the maximum displacement is equal to the amplitude (A), which is given as 5 mm or 0.005 m. Plugging in the values, we have:

a = (40π)^2 * 0.005
a ≈ 2503 m/s^2

Therefore, the acceleration at maximum displacement is approximately 2503 m/s^2.

To find the maximum kinetic energy (Ek) of the mass, we can use the formula:

Ek = (1/2) * m * v^2

Where:
- Ek is the kinetic energy
- m is the mass
- v is the velocity

In this case, the mass (m) is given as 2 g, which is equivalent to 0.002 kg. Plugging in the values, we have:

Ek = (1/2) * 0.002 * (0.628)^2
Ek ≈ 0.000394 J

Therefore, the maximum kinetic energy (Ek) of the mass is approximately 0.000394 J.

Answer 3

To find the velocity of the end when passing through the zero position, we can use the equation for simple harmonic motion (SHM):

v = ω * A

Where:
v is the velocity of the end,
ω is the angular frequency,
A is the amplitude of vibration.

First, we need to find ω. The angular frequency is related to the frequency of vibration (f) by the formula:

ω = 2πf

Given that the frequency is 20 Hz, we can substitute this value into the formula to find ω:

ω = 2π * 20
= 40π rad/s

Next, we can substitute the values of ω and A into the formula for velocity:

v = 40π * 5mm
≈ 628.32 mm/s

Therefore, the velocity of the end when passing through the zero position is approximately 628.32 mm/s.

To find the acceleration at maximum displacement, we can use the formula:

a = -ω^2 * x

Where:
a is the acceleration,
ω is the angular frequency,
x is the maximum displacement.

Given that the maximum displacement (x) is equal to the amplitude (A), which is 5mm, we can substitute the values into the formula:

a = - (40π)^2 * 5mm
≈ - 7907.98 mm/s^2

Therefore, the acceleration at maximum displacement is approximately -7907.98 mm/s^2, indicating that the acceleration is directed opposite to the displacement.

Finally, to find the maximum kinetic energy (Ek) of the mass, we need to use the formula:

Ek = (1/2) * m * v^2

Where:
Ek is the maximum kinetic energy,
m is the mass, and
v is the velocity of the end.

Given that the mass is 2g, we need to convert it to kilograms by dividing by 1000:

m = 2g ÷ 1000
= 0.002 kg

Now, we can substitute the values into the formula:

Ek = (1/2) * 0.002 kg * (628.32 mm/s)^2
≈ 0.394 J

Therefore, the maximum kinetic energy of the mass is approximately 0.394 Joules.