Movement is one of the important properties of matter in our Universe. Indeed, even at absolute zero temperatures, the movement of particles of matter does not stop completely. In physics, motion is described by a number of parameters, the main of which is acceleration. In this article, we will reveal in more detail the question of what constitutes tangential acceleration and how to calculate it.
Acceleration in physics
Under the acceleration understand the speed with which the speed of the body changes during its movement. Mathematically, this definition is written as follows:
a¯=d v¯/ d t
This is the kinematic definition of acceleration. The formula shows that it is calculated in meters per square second (m/s2). Acceleration is a vector characteristic. Its direction has nothing to do with the direction of speed. Directed acceleration in the direction of speed change. Obviously, in the case of uniform motion in a straight line, there is nono change in speed, so acceleration is zero.
If we talk about acceleration as a quantity of dynamics, then we should remember Newton's law:
F¯=m × a¯=>
a¯=F¯ / m
The cause of the quantity a¯ is the force F¯ acting on the body. Since the mass m is a scalar value, the acceleration is directed in the direction of the force.
Trajectory and full acceleration
Speaking of acceleration, speed and the distance traveled, one should not forget about another important characteristic of any movement - the trajectory. It is understood as an imaginary line along which the studied body moves. In general, it can be curved or straight. The most common curved path is the circle.
Assume that the body moves along a curved path. At the same time, its speed changes according to a certain law v=v (t). At any point of the trajectory, the velocity is directed tangentially to it. The speed can be expressed as the product of its modulus v and the elementary vector u¯. Then for acceleration we get:
v¯=v × u¯;
a¯=d v¯/ d t=d (v × u¯) / d t
Applying the rule for calculating the derivative of the product of functions, we get:
a¯=d (v × u¯) / d t=d v / d t × u¯ + v × d u¯ / d t
Thus, the total acceleration a¯ when moving along a curved pathis decomposed into two components. In this article, we will consider in detail only the first term, which is called the tangential acceleration of a point. As for the second term, let's just say that it is called normal acceleration and is directed towards the center of curvature.
Tangential acceleration
Let's designate this component of total acceleration as at¯. Let's write down the formula for tangential acceleration again:
at¯=d v / d t × u¯
What does this equality say? First, the component at¯ characterizes the change in the absolute value of the speed, without taking into account its direction. So, in the process of movement, the velocity vector can be constant (rectilinear) or constantly change (curvilinear), but if the velocity modulus remains unchanged, then at¯ will be equal to zero.
Secondly, the tangential acceleration is directed exactly the same as the velocity vector. This fact is confirmed by the presence in the formula written above of a factor in the form of an elementary vector u¯. Since u¯ is directed tangentially to the trajectory, the component at¯ is often referred to as tangential acceleration.
Based on the definition of tangential acceleration, we can conclude: the values a¯ and at¯ always coincide in the case of rectilinear movement of the body.
Tangential and angular acceleration when moving in a circle
Above we found outthat the movement along any curvilinear trajectory leads to the appearance of two components of acceleration. One of the types of movement along a curved line is the rotation of bodies and material points along a circle. This type of movement is conveniently described by angular characteristics, such as angular acceleration, angular velocity and angle of rotation.
Under the angular acceleration α understand the magnitude of the change in the speed of the angular ω:
α=d ω / d t
Angular acceleration leads to an increase in rotational speed. Obviously, this increases the linear velocity of each point that participates in the rotation. Therefore, there must be an expression that relates the angular and tangential acceleration. We will not go into the details of the derivation of this expression, but we will give it right away:
at=α × r
The values at and α are directly proportional to each other. In addition, at increases with increasing distance r from the rotation axis to the considered point. That is why it is convenient to use α during rotation, and not at (α does not depend on the rotation radius r).
Example problem
It is known that a material point rotates around an axis with a radius of 0.5 meters. Its angular velocity in this case changes according to the following law:
ω=4 × t + t2+ 3
It is necessary to determine with what tangential acceleration the point will rotate at time 3.5 seconds.
To solve this problem, you should first use the formula for the angular acceleration. We have:
α=d ω/ d t=2 × t + 4
Now you should apply the equality that relates the quantities at and α, we get:
at=α × r=t + 2
When writing the last expression, we substituted the value r=0.5 m from the condition. As a result, we have obtained a formula according to which tangential acceleration depends on time. Such circular motion is not uniformly accelerated. To obtain an answer to the problem, it remains to substitute a known point in time. We get the answer: at=5.5 m/s2.
