A body of mass 10 kg is moving over a smooth surface, whose equation of motion is given by the relation. s = 5t + 2t² where (s) is in metres and (t) in seconds. Find the magnitude of force responsible for the motion. ? Step by step
P help me.
a = s" = 4
F = ma
Sure, I'd be happy to help you step by step! To find the magnitude of the force responsible for the motion, we can start by finding the second derivative of the equation of motion.
Given the equation of motion:
s = 5t + 2t²
First, let's find the first derivative of s with respect to t:
ds/dt = 5 + 4t
Next, let's find the second derivative of s with respect to t (which will give us the acceleration):
d²s/dt² = 4
Now, we have the acceleration of the object, which is 4 m/s².
Using Newton's second law of motion, which states that force (F) equals mass (m) times acceleration (a):
F = m * a
Let's substitute the given mass of 10 kg and the acceleration we found:
F = 10 kg * 4 m/s² = 40 N
Therefore, the magnitude of the force responsible for the motion is 40 Newtons.
To find the magnitude of the force responsible for the motion, we need to calculate the second derivative of the equation of motion and then multiply it by the mass of the body.
Step 1: Calculate the first derivative of the equation of motion:
ds/dt = 5 + 4t
Step 2: Calculate the second derivative of the equation of motion:
d²s/dt² = d/dt(5 + 4t) = 4
Step 3: Multiply the second derivative by the mass of the body:
Force = mass * acceleration
Force = 10 kg * 4 m/s²
Step 4: Calculate the magnitude of the force:
Force = 40 N
Therefore, the magnitude of the force responsible for the motion is 40 N.
To find the magnitude of the force responsible for the motion, we need to differentiate the equation of motion with respect to time (t) to obtain the acceleration (a).
Given: s = 5t + 2t^2
Step 1: Differentiate the equation of motion with respect to time (t) to find the acceleration (a).
ds/dt = d/dt(5t + 2t^2)
The derivative of 5t with respect to t is simply 5.
The derivative of 2t^2 with respect to t can be found using the power rule. Multiply the coefficient (2) by the exponent (2), and then subtract 1 from the exponent to get 4t.
Therefore, ds/dt = 5 + 4t.
Step 2: Calculate the magnitude of acceleration by substituting the given time (t) value into the equation above.
Let's assume the time (t) at which we want to find the magnitude of the force is t1.
At t = t1, ds/dt = 5 + 4t1.
Step 3: Use Newton's second law of motion, which states that force (F) is equal to mass (m) multiplied by acceleration (a):
F = m * a
Given: mass (m) = 10 kg.
Substituting the calculated acceleration (a) into the equation, we have:
F = 10 * (5 + 4t1)
Therefore, the magnitude of the force responsible for the motion is 10 times the quantity of (5 + 4t1).