# Body on a spring: expression for the period T

• moenste
In summary, the period T of vertical oscillations for a body of mass m suspended from a vertical, light, helical spring of force constant k is given by T = 2 π √(m / k). When two identical springs are joined in series, the period becomes T = 2 π √(m / (k / 2)), and when placed in parallel, the period is T = 2 π √(m / (2k)). The only change is in the value of k due to the different spring positions.
moenste

## Homework Statement

(a) A body of mass m is suspended from a vertical, light, helical spring of force constant k, as in Fig. 1. Write down an expression for the period T of vertical oscillations of m.

(b) Two such identical springs are now joined as in Fig. 2 and support the same mass m. In terms of T, what is the period of vertical oscillations in this case?

(c) The identical springs are now placed side by side as in Fig. 3, and m is supported symmetrically from them by means of a weightless bar. In terms of T, what is the period of vertical oscillations in this case?

2. The attempt at a solution
(a) The first figure looks like a basic example of a body on a spring and it's period should be: T = 2 π √(m / k).

(b) Since there are two identical springs, I would say that their force constant k should be less, because the spring is longer now: T = 2 π √(m / (k / 2)).

(c) Since the two springs are in parallel, the force constant k should be twice as much: T = 2 π √(m / (2k)).

I am not sure for the (b) and (c) parts. If we take the (a) formula as the basis, the only thing (as I see it) that changes is k due to the different spring positions. So, for (b) it should be (k / 2) and for (c) 2k. I am going in the right direction?

moenste said:
If we take the (a) formula as the basis, the only thing (as I see it) that changes is k due to the different spring positions. So, for (b) it should be (k / 2) and for (c) 2k. I am going in the right direction?
Exactly right!

moenste

## 1. What is the equation for the period of a body on a spring?

The equation for the period (T) of a body on a spring is T = 2π√(m/k), where m is the mass of the object and k is the spring constant.

## 2. How does the mass of the object affect the period of a body on a spring?

The period of a body on a spring is directly proportional to the square root of the mass. This means that as the mass increases, the period also increases.

## 3. How does the spring constant affect the period of a body on a spring?

The period of a body on a spring is inversely proportional to the square root of the spring constant. This means that as the spring constant increases, the period decreases.

## 4. Can the period of a body on a spring be changed?

Yes, the period of a body on a spring can be changed by altering the mass of the object or the spring constant. Other factors such as air resistance and damping also affect the period.

## 5. How is the motion of a body on a spring affected by the period?

The period of a body on a spring determines the frequency of its oscillations. A shorter period results in a higher frequency and a faster motion, while a longer period results in a lower frequency and a slower motion.

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