Question

A mass m is oscillating with amplitude A at the end of a vertical spring of spring constant k. The mass is increased by a factor of four.1. What is true about the period? (period does what factor)2. The mass is increased by a factor of four, keeping A and k constant. What is true about the total mechanical energy?3. The mass is increased by a factor of four, keeping A and k constant. What is true about the maximum speed?4. The amplitude is doubled, keeping mass and k constant. What is true about the period?5. The amplitude is doubled, keeping mass and k constant. What is true about the total mechanical energy? Choose the best answer.6. The amplitude is doubled, keeping mass and k constant. What is true about the maximum speed?

Answer 1

Answer:

1. The period doubles

2. The mechanical energy is unchanged

3. The speed is halved

4. The period is unchanged

5. The energy is quadrupled

6. The maximum is speed doubled

Explanation:

1.  From Hooke's law we have $T\propto\sqrt{\frac{m}{k} }$

where T is period, m is mass and k is the spring constant

So If the mass is increased by a factor of 4 then period doubles while k constant

2 According to law of conservation of energy, the energy remains unchanged so therefore the total mechanical energy remain unchanged.

3. From the  $w = \sqrt{\frac{k}{m} }$

where m is  mass and w is speed

so we see that mass and speed is inversely proportional therefore if we increase mass speed decreases.

4. The period is independent of amplitude

5. $E=A^{2}$

where E is energy and A is amplitude

So if the amplitude is doubled the energy is quadrupled.

6.  we have the relation

$v_{max}=A$

where $v_{max}$ is maximum speed and A is amplitude  

From the formular we can see they are directly proportional, so if we double amplitude then $v_{max}$ doubles also.