A block of mass m and charge q placed on a smooth horizontal surface connected through an unstretched spring fixed at one end, of spring constant k. An electric field parallel to t... A block of mass m and charge q placed on a smooth horizontal surface connected through an unstretched spring fixed at one end, of spring constant k. An electric field parallel to the spring is applied, the resulting amplitude of SHM in the term of x(qE/k). Find the value of x?
Understand the Problem
The question is asking to determine the value of x in the context of simple harmonic motion (SHM) of a block connected to a spring, influenced by an electric field. It involves physics concepts related to mass, charge, and spring constant.
Answer
The value of $x$ is $1$.
Answer for screen readers
The value of $x$ is 1.
Steps to Solve
- Identify the forces acting on the block
The block experiences two main forces: the restoring force of the spring and the electric force due to the electric field. The restoring force provided by the spring is given by Hooke’s law:
$$ F_{spring} = -kx $$
where $k$ is the spring constant and $x$ is the displacement from the equilibrium position.
- Calculate the electric force
The electric force ($F_{electric}$) on the charged block in an electric field $E$ is given by:
$$ F_{electric} = qE $$
where $q$ is the charge on the block.
- Set up the equilibrium condition
At equilibrium, the net force acting on the block is zero. Therefore, we can equate the spring force and the electric force:
$$ kx = qE $$
- Solve for displacement $x$
Rearranging the equation gives us:
$$ x = \frac{qE}{k} $$
- Determine the amplitude of SHM in terms of $x$
Since the amplitude of SHM ($A$) will be equal to the maximum displacement ($x$), we can state that:
$$ A = \frac{qE}{k} $$
- Express in the required format
According to the question, we need to express the amplitude in terms of $\frac{qE}{k}$. Thus, we can write:
$$ A = x \cdot \frac{qE}{k} $$
From this step, we identify that the coefficient $x$ corresponds to the ratio given.
The value of $x$ is 1.
More Information
In this problem, we examined the forces acting on a block in a simple harmonic motion scenario. The displacement due to the electric force and the spring force allowed us to find the relationship between these forces, leading to identifying the amplitude of SHM.
Tips
- Ignoring the direction of forces: Always consider the direction of forces when setting up equilibrium equations.
- Confusing the role of displacement and amplitude: Ensure that the maximum displacement corresponds to the amplitude in SHM.