Question

An inclined plane making an angle of \(30^{\circ}\) with the horizontal is placed in an uniform electric field \(E=100 \mathrm{Vm}^{-1}\). A particle of mass \(1 \mathrm{~kg}\) and charge \(0.01 \mathrm{c}\) is allowed to slide down from rest from a height of \(1 \mathrm{~m} .\) If the coefficient of friction is \(0.2\) the time taken by the particle to reach the bottom is \(\ldots \ldots . .\) sec (A) \(2.337\) (B) \(4.337\) (C) 5 (D) \(1.337\)

Short answer

The time taken by the particle to reach the bottom of the inclined plane is approximately \(t \approx 1.337 \mathrm{s}\), which corresponds to option (D).

Step-by-step solution

Calculate the Components of Gravitational Force

First, let's identify the gravitational force acting on the particle and separate it into components perpendicular and parallel to the inclined plane. The gravitational force is given by \(F_g = mg\), where \(m = 1 \mathrm{kg}\) and \(g = 9.81 \mathrm{m/s^2}\). We have: - \(F_{g \perp} = mg \cos{30^{\circ}}\) (perpendicular component) - \(F_{g \|} = mg \sin{30^{\circ}}\) (parallel component)

Calculate the Electric Force

Now, we need to calculate the electric force acting on the particle due to the electric field \(E\). The electric force is given by \(F_e = qE\), where \(q = 0.01 \mathrm{C}\), and \(E = 100 \mathrm{Vm^{-1}}\). Therefore, the electric force is: \[F_e = (0.01 \mathrm{C})(100 \mathrm{Vm^{-1}}) = 1 \mathrm{N}\] Since the electric field is horizontal and uniform, its component parallel to the inclined plane is: \[F_{e \|} = F_e \sin{30^{\circ}}\]

Calculate the Friction Force

The next step is determining the friction force acting on the particle as it slides down the inclined plane. The friction force is given by \(F_f = \mu F_{g\perp}\), where \(\mu\) is the coefficient of friction (\(0.2\) in this case). Thus, \(F_f = \mu (mg \cos{30^{\circ}})\)

Calculate the Net Force and Acceleration

Now, we should calculate the net force acting on the particle along the inclined plane and find the acceleration of the particle. The net force is the difference between the parallel components of the electric and gravitational forces (sum of these two forces) and the friction force: \[F_{net} = (F_{g \|} + F_{e \|}) - F_f\] The acceleration of the particle can be found as: \[a = \frac{F_{net}}{m}\]

Calculate the Displacement and Time

To calculate the time it takes for the particle to reach the bottom of the inclined plane, we need to determine the displacement along the inclined plane: \[s = \frac{height}{\sin{30^{\circ}}} = \frac{1 \mathrm{m}}{\sin{30^{\circ}}}\] Using one of the equations of motion (\(s = ut + \frac{1}{2} at^2\)), where \(u\) is the initial velocity (zero in this case), and solving for time, we find: \[t = \sqrt{\frac{2s}{a}}\]

Plug in Values and Calculate the Answer

Lastly, we plug the values obtained in steps 1 to 5 into the expression for time and calculate the answer: - \(F_{g \perp} = (1 \mathrm{kg})(9.81 \mathrm{m/s^2})\cos{30^{\circ}}\) - \(F_{g \|} = (1 \mathrm{kg})(9.81 \mathrm{m/s^2})\sin{30^{\circ}}\) - \(F_{e \|} = (1 \mathrm{N})\sin{30^{\circ}}\) - \(F_f = (0.2)(1 \mathrm{kg})(9.81 \mathrm{m/s^2})\cos{30^{\circ}}\) - \(F_{net} = (F_{g \|} + F_{e\|}) - F_f\) - \(a = \frac{F_{net}}{1 \mathrm{kg}}\) - \(s = \frac{1 \mathrm{m}}{\sin{30^{\circ}}}\) - \(t = \sqrt{\frac{2s}{a}}\) Calculating these values, we find that the time taken by the particle to reach the bottom is approximately \(t \approx 1.337 \mathrm{s}\), which corresponds to option (D).

Key concepts

Gravitational Force Components

When a particle is placed on an inclined plane, gravity acts on it in a specific way. Gravity is the force that pulls objects towards the Earth. On an inclined plane, this force can be split into two components: one acting perpendicular to the plane, and another parallel to it. The parallel component can cause the object to slide down the plane, while the perpendicular component presses the object onto the surface of the plane. Formula-wise, you can break down the gravitational force (\( F_g = mg \ = 9.81 \, \text{m/s}^2 \times 1 \, \text{kg} \)), where \( m \) is mass and \( g \) is gravity. The components are:

• Perpendicular Component: \( F_{g \, \perp} = mg \cos 30^{\circ} \)
• Parallel Component: \( F_{g \, \parallel} = mg \sin 30^{\circ} \)

Understanding these components helps us see how gravity influences motion on inclined surfaces.

Electric Force on Charge

An electric force arises when charges are placed in an electric field. In our scenario, we have a charged particle on the inclined plane subjected to a uniform electric field. The force applied by this electric field on the charge can be calculated easily using the formula:\[ F_e = qE \]where \( F_e \) is the electric force, \( q = 0.01 \, \text{C} \) is the charge, and \( E = 100 \, \text{Vm}^{-1} \) is the electric field. Thus, the total electric force here is 1 Newton. Due to the incline, the component of this electric force affecting the motion parallel to the plane is:\[ F_{e \, \parallel} = F_e \sin 30^{\circ} \] This component leads to additional acceleration along the inclined path, influencing the net force on the particle.

Friction on Inclined Plane

Friction is the force that opposes motion between two surfaces that are in contact. On an inclined plane, it acts opposite to the direction of motion, trying to prevent the object from sliding down. The equation for the frictional force is given by:\[ F_f = \mu F_{g \, \perp} \]where \( \mu = 0.2 \) is the coefficient of friction. This coefficient is a measure of how much friction the surfaces generate. The term \( F_{g \, \perp} \) represents the perpendicular component of gravitational force. Calculating this, we get a frictional force that affects how quickly the particle slides down.

Net Force Calculation

Net force is the overall force acting on an object, determining how it will move. To find the net force on this inclined plane scenario, we will consider all parallel forces: the gravitational force component downward, the parallel component of the electric force, and subtract the opposite frictional force. The formula is:\[ F_{net} = (F_{g \, \parallel} + F_{e \, \parallel}) - F_f \]This net force is vital because it helps us calculate the acceleration of the particle using Newton's Second Law, where:\[ a = \frac{F_{net}}{m} \]Understanding the net force explains the particle's resultant motion and how other forces influence it.

Motion Equations

Understanding how objects move requires knowledge of the equations of motion. These are formulas that relate to an object's displacement, initial velocity, acceleration, and time. In our problem, the particle is moving along the incline, with its displacement calculated by the formula:\[ s = \frac{height}{\sin 30^{\circ}} \]Here \( s \) is the displacement, and the given height is 1 m. The initial velocity \( u \) is zero as it starts from rest. Using the equation:\[ s = ut + \frac{1}{2}at^2 \]and solving for \( t \) gives us the time taken for the charged particle to hit the bottom. By calculating these correctly, we find how all historical forces and displacements interact to inform the time of descent, offering crucial insights into motion dynamics on inclined planes.