(a) Find an expression for the distance d the spring is stretched from equilibrium. (Use any variable or symbol stated above along with the following as necessary: k and g.) d = (b) Find expressions for the components of the force exerted by the pivot on the beam. (Use the following as necessary: m, k, g, and 0. Assume the positive x-direction is to the right and the positive y-direction is upward.) Rx =
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
The figure shows a uniform beam of mass m pivoted at its lower end, with a horizontal spring attached between its top end and a vertical wall. The beam makes an angle ? with the horizontal. Answer parts a-b.
The system consists of the following elements:
- A spring with spring constant \( k \)
- A beam with mass \( m \)
- An angle \( \theta \)
- Gravitational constant \( g \)
The spring is attached to a fixed surface on one end and the beam on the other end. The beam pivots about a point at its base.
#### Task (a):
Find an expression for the distance \( d \) the spring is stretched from equilibrium. (Use any variable or symbol stated above along with the following as necessary: \( k \) and \( g \)).
\[ \text{Expression: } d = \, \_\_\_\_\_\_\_\_\_\_\_\_ \]
#### Task (b):
Find expressions for the components of the force exerted by the pivot on the beam. (Use the following as necessary: \( m \), \( k \), \( g \), and \( \theta \)). Assume the positive \( x \)-direction is to the right and the positive \( y \)-direction is upward.
1. Horizontal Component (\( R_x \)):
\[ R_x = \, \_\_\_\_\_\_\_\_\_\_\_\_ \]
2. Vertical Component (\( R_y \)):
\[ R_y = \, \_\_\_\_\_\_\_\_\_\_\_\_ \]
#### Diagram Explanation:
The diagram illustrates the following elements and forces in the system:
- The spring constant \( k \)
- The mass of the beam \( m \)
- The angle \( \theta \) that the beam makes with the horizontal or vertical axis
- The gravitational force \( g \)
This is a statics and dynamics problem involving equilibrium conditions for the spring and the beam system. The goal is to determine the displacement of the spring due to the weight of the beam and the force components exerted by the pivot.
Note: The detailed solved expressions for \( d \), \( R_x \), and \( R_y \) require the application of physical principles including Hooke’s law for springs and Newton’s laws of motion for the beam's equilibrium.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86fc9270-ad68-4ba8-8b0d-981a01802a8a%2F0f2e6808-2585-439f-9a49-99cff396b962%2Ff5tp0z_processed.png&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 4 images
