Uniform Circular Motion Formula: Circular motion refers to the movement along a curved path. When an object covers an equal distance along the circumference in the same interval of time, it undergoes uniform circular motion. In this type of motion, the speed of the object remains constant, but the direction of its velocity changes. Throughout uniform circular motion, the tangential speed at any point on the circumference remains constant.
The tangential velocity vector is always tangent to the circumference at each point. Tangential velocity is a vector aligned with tangential speed, maintaining a constant magnitude equal to the tangential speed of the motion. The formula for tangential speed ( v) in uniform circular motion is given by:
v= 2πr / T
where:
v is the tangential velocity,
r is the radius,
T is the period (time required to complete one full circle).
Centripetal acceleration ( a) in uniform circular motion is determined by the formula:
a= v
2
/r
where: v is the tangential velocity, r is the radius.
In uniform circular motion, an object also possesses angular velocity. The velocity of the object undergoes continuous change, resulting in an acceleration in the circular motion.
This acceleration, known as centripetal acceleration, is always directed towards the center of the circle.
Uniform Circular Motion Formula Solved Examples
Example 1 :
A satellite in Earth's orbit has a tangential speed of 10,000 m/s. If the radius of its circular orbit is 7,000 kilometers, calculate the centripetal acceleration.
Solution: Given: Tangential speed ( v) = 10,000 m/s Radius ( r) = 7,000,000 meters (converted from 7,000 kilometers)
We can use the formula for centripetal acceleration: a= v
2
/r
Substituting the values, we get:
a= (10,000)
2
/ 7,000,000
Simplifying:
a= 100,000,000 / 7,000,000
a≈14.29m/s
2
Therefore, the centripetal acceleration of the satellite is approximately 14.29m/s
2
.
Example 2:
A wheel with a radius of 0.5 meters is rotating with an angular velocity of 4 radians per second. Determine the tangential speed of a point on the edge of the wheel.
Solution: Given: Radius ( r) = 0.5 meters Angular velocity ( ω) = 4 radians/second The tangential speed ( v) is given by the formula:
v=r⋅ω
Substituting the values, we find:
v=0.5⋅4
v=2m/s
Hence, the tangential speed of a point on the edge of the wheel is 2m/s.
Example 3:
A car is moving around a circular track with a radius of 200 meters. The car's tangential speed is 25 m/s. Calculate the period of its motion.
Solution: Given: Radius ( r) = 200 meters Tangential speed ( v) = 25 m/s We can use the formula for the period ( T) in circular motion:
T= 2πr /v Substituting the values, we get:
T= 2π×200 /25
Simplifying: T= 400π /25
T=16π
Therefore, the period of the car's motion is 16π seconds.
The study of circular motion involves understanding various concepts such as tangential speed, tangential velocity, centripetal acceleration, and angular velocity. These concepts are crucial in describing the motion of objects moving along curved paths or in circular orbits. Key formulas for calculating these quantities include:
Tangential Speed (v):
v= 2πr/T
where v is the tangential velocity, r is the radius, and T is the period.
Centripetal Acceleration ( a):
a= v
2
/r
where v is the tangential velocity, and r is the radius.
Tangential Velocity ( v):
v=r⋅ω
where r is the radius, and ω is the angular velocity.
These formulas provide a quantitative understanding of the dynamics involved in circular motion. Applying these principles to real-world scenarios allows us to calculate important parameters such as acceleration, speed, and angular velocity, providing valuable insights into the behavior of objects in circular motion.
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