# Simple Harmonic Motion, masses, and springs.

• csnsc14320
In summary, two blocks (m = 1.22 kg and M = 8.73 kg) and a spring (k = 344 N/m) are arranged on a horizontal, frictionless surface with m resting on top of M and the spring attached to a wall and M. The maximum possible amplitude of the simple harmonic motion is found by setting the equations for the acceleration of both blocks equal to each other and solving for x. The correct expression is x = (M + m)/(k*μs*g), and this was confirmed by checking units. The acceleration is greatest at the maximum displacement, x, and it is a function of time. Therefore, to find the maximum acceleration, the position function must be differentiated twice
csnsc14320

## Homework Statement

Two blocks (m = 1.22 kg and M = 8.73 kg) and a spring (k = 344 N/m) are arranged on a horizontal, frictionless surface, with m resting on top of M and the spring attached to a wall and M. The coefficient of static friction between the blocks is 0.42. Find the maximum possible amplitude of the simple harmonic motion is no slippage is to occur between the blocks.

## The Attempt at a Solution

So, after drawing free body diagrams for both blocks, I get:

m: $$\sum F = -\mu_s m g = m a$$ or just $$a = -\mu_s g$$

M: $$-k x = (M + m) a$$ or just $$a = -\frac{M + m}{k x}$$

Setting these two equations equal, I finally get:

$$x = \frac{M + m}{k \mu_S g}$$

However, when I plug in the values, I get the wrong answer. I think something may be wrong with my FBD's?

Sunav K Vidhyarthi
The acceleration is a function of time. Diffentiate the position function twice with respect to time and find the maximum acceleration. This expression will give the acceleration as a function of omega and amplitude.

Wouldn't the acceleration be the greatest at the maximum displacement? That's why I thought it would be find to only consider the acceleration at the point of maximum amplitude, x.

Also, I have two equations for the acceleration? which would I differentiate? Or does it matter?

csnsc14320 said:
M: $$-k x = (M + m) a$$
OK.
or just $$a = -\frac{M + m}{k x}$$
Oops... redo this step. (You would have caught this if you had checked units.)

Doc Al said:

OK.

Oops... redo this step. (You would have caught this if you had checked units.)

Oops! I need to slow down a little bit when doing my simple algebra

I re-did it, checked my units (came out to meters),plugged it, and got the right answer.

Thanks

## 1. What is simple harmonic motion?

Simple harmonic motion is a type of periodic motion in which a system moves back and forth around an equilibrium point with a constant amplitude and a constant period. It is often seen in systems involving masses and springs.

## 2. How is simple harmonic motion related to masses and springs?

Simple harmonic motion is commonly observed in systems involving masses and springs. When a mass is attached to a spring and pulled from its equilibrium position, the spring exerts a force on the mass that causes it to oscillate back and forth. This creates a repetitive pattern known as simple harmonic motion.

## 3. What is the role of mass in simple harmonic motion?

Mass is an important factor in simple harmonic motion because it determines the inertia of the system. A larger mass will have a greater inertia, meaning it will require a greater force to move it from its equilibrium position. This affects the period and frequency of the motion.

## 4. How does the spring constant affect simple harmonic motion?

The spring constant, also known as the stiffness of the spring, affects the force applied to the mass and therefore the amplitude and period of the motion. A higher spring constant will result in a stiffer spring, which will cause the mass to oscillate more quickly and with a smaller amplitude.

## 5. Can simple harmonic motion occur without the use of a spring?

Yes, simple harmonic motion can occur without the use of a spring. It can also be seen in systems such as pendulums, where a mass is attached to a string and swings back and forth around a fixed point. In these cases, the restoring force is provided by gravity instead of a spring.

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