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This article aims to find acceleration imparted on the box placed on a frictionless table on earth.
In mechanics, acceleration is rate of change of an object’s velocity with respect to time. Accelerations are vector quantities having both magnitude and direction. The direction of an object’s acceleration is given by orientation of the net force acting on that object. The magnitude of the object’s acceleration, as described by Newton’s second law, is the combined effect of two causes:
- The net balance of all external forces acting on that object — the magnitude is directly proportional to this resulting resultant force
- The weight of that object, depending on the materials it is made of — size is inversely proportional to the object’s mass.
The SI unit is meters per second squared, $\dfrac{m}{s^{2}}$.
Average Acceleration
Average acceleration
Average acceleration is the rate of change of velocity $\Delta v$ divided over the time $\Delta t$.
\[a=\dfrac{\Delta v}{\Delta t}\]
Instantaneous Acceleration
Instantaneous acceleration is the limit of average acceleration over an infinitesimally small time interval. Numerically, the instantaneous acceleration is the derivative of the velocity vector with respect to time.
\[a=\dfrac{dv}{dt}\]
Since acceleration is defined as the derivative of velocity $v$ with respect to time $t$ and velocity are defined as derivative of position $x$ with respect to time, acceleration can be thought of as second derivative of $x$ with respect to $t$:
\[a=\dfrac{dv}{dt}=\dfrac{d^{2}x}{d^{2}t}\]
Newton’s Second Law of Motion
The proper acceleration, i.e., the acceleration of the body relative to the state of free fall, is measured by an accelerometer. In classical mechanics, for a body having constant mass (vector), the acceleration of the body’s center of gravity is proportional to the net force vector (i.e., the sum of all forces) acting on it (Newton’s second law):
\[F=ma\]
\[a=\dfrac{F}{m}\]
$F$ is the net force acting on body, and $m$ is the mass.
Mass
Newton 2nd law
Expert Answer
Data given in the question is:
\[a(acceleration) of \: the \:block=5.3\dfrac{m}{s^{2}}\]
\[F(horizontal force)=10\:N\]
\[a(acceleration)\: due \:to\:gravity=1.62\dfrac{m}{s^{2}}\]
The value of mass is calculated by using the following formula:
\[F=\dfrac{m}{a}\]
\[m=\dfrac{F}{a}\]
\[m=\dfrac{10}{5.3}\]
\[m=1.89\:kg\]
The mass of the box is $1.89\:kg$.
The value of the acceleration is found by using the following formula:
\[F=ma\]
\[a=\dfrac{F}{m}\]
\[a=\dfrac{5}{1.89}\]
\[a=2.65\dfrac{m}{s^{2}}\]
Hence, acceleration imparted to the block is $2.65\dfrac{m}{s^{2}}$.
Numerical Result
Acceleration imparted to the block is $2.65\dfrac{m}{s^{2}}$.
Example
The block is on a frictionless table on the ground. The block accelerates at $5\dfrac{m}{s^{2}}$ when acted upon by a horizontal force of $20\: N$. The block and table are placed on the moon. Gravitational acceleration on the surface of the Moon is $1.8\dfrac{m}{s^{2}}$.When the block is on the moon, a horizontal force of $15\:N$ acts on it.
Solution
Data given in the example is:
\[a(acceleration) of \: the \:block=5\dfrac{m}{s^{2}}\]
\[F(horizontal force)=20\:N\]
\[a(acceleration)\: due \:to\:gravity=1.8\dfrac{m}{s^{2}}\]
The value of mass is calculated by using the following formula:
\[F=\dfrac{m}{a}\]
\[m=\dfrac{F}{a}\]
\[m=\dfrac{20}{5}\]
\[m=4\:kg\]
The mass of the box is $4\:kg$.
The value of the acceleration is found by using the following formula:
\[F=ma\]
\[a=\dfrac{F}{m}\]
\[a=\dfrac{15}{4}\]
\[a=3.75\dfrac{m}{s^{2}}\]
Hence, acceleration imparted to the block is $3.75\dfrac{m}{s^{2}}$.