A car is moving in a circular horizontal track of radius 10 m with a constant speed of 10 m/s-1. A bob is suspended from the roof of the car by a light wire of length 1.0 m. The angle made by the wire with the vertical is (in radian)
A car is moving in a circular horizontal track of radius 10 m with a constant speed of 10 m/s-1. A bob is suspended from the roof of the car by a light wire of length 1.0 m. The angle made by the wire with the vertical is (in radian)
- 0
- π/6
- π/3
- π/4
The Correct Option is D
Solution and Explanation
At any point on the circular track, the bob is subject to two forces: The tension force in the wire, acting towards the center of the circle. The gravitational force, acting vertically downward.
Since the car is moving with a constant speed, the bob is in equilibrium, and the net force acting on it must be zero. Considering the forces in the vertical direction, we have:
Tension force (T) * cos(θ) - Weight (mg) = 0
Considering the forces in the horizontal direction, we have:
Tension force (T) * sin(θ) = Centripetal force (mv²/r)
We can now solve these equations to find the angle θ.
From the equation T * cos(θ) - mg = 0, we have T = mg / cos(θ).
Substituting this value of T into the equation T * sin(θ) = mv²/r, we get:
(mg / cos(θ)) * sin(θ) = mv²/r
Simplifying, we have: tan(θ) = v² / (rg)
Given that v = 10 m/s and r = 10 m, we can calculate the value of tan(θ):
tan(θ) = (10²) / (10 * 10) = 1
To find the angle θ, we take the inverse tangent (arctan) of both sides:
θ = arctan(1)
θ = π/4
Therefore, the angle made by the wire with the vertical is π/4 radians (option D).
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Concepts Used:
Uniform Circular Motion
A circular motion is defined as the movement of a body that follows a circular route. The motion of a body going at a constant speed along a circular path is known as uniform circular motion. The velocity varies while the speed of the body in uniform circular motion remains constant.
Uniform Circular Motion Examples:
- The motion of electrons around its nucleus.
- The motion of blades of the windmills.
Uniform Circular Motion Formula:
When the radius of the circular path is R, and the magnitude of the velocity of the object is V. Then, the radial acceleration of the object is:
arad = v2/R
Similarly, this radial acceleration is always perpendicular to the velocity direction. Its SI unit is m2s−2.
The radial acceleration can be mathematically written using the period of the motion i.e. T. This period T is the volume of time taken to complete a revolution. Its unit is measurable in seconds.
When angular velocity changes in a unit of time, it is a radial acceleration.
Angular acceleration indicates the time rate of change of angular velocity and is usually denoted by α and is expressed in radians per second. Moreover, the angular acceleration is constant and does not depend on the time variable as it varies linearly with time. Angular Acceleration is also called Rotational Acceleration.
Angular acceleration is a vector quantity, meaning it has magnitude and direction. The direction of angular acceleration is perpendicular to the plane of rotation.
Formula Of Angular Acceleration
The formula of angular acceleration can be given in three different ways.
α = dωdt
Where,
ω → Angular speed
t → Time
α = d2θdt2
Where,
θ → Angle of rotation
t → Time
Average angular acceleration can be calculated by the formula below. This formula comes in handy when angular acceleration is not constant and changes with time.
αavg = ω2 - ω1t2 - t1
Where,
ω1 → Initial angular speed
ω2 → Final angular speed
t1 → Starting time
t2 → Ending time
Also Read: Angular Motion
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