The displacement of a particle with a mass of 225 g depends on time and is described by the function x(t) = -3.0t3 + 2.0t2 + 4.0t – 13. All constants are in SI units. Calculate the resultant force acting on the particle at time 2.3 s.
To find the resultant force acting on the particle, we need to take the second derivative of the displacement function with respect to time:
x(t) = -3.0t3 + 2.0t2 + 4.0t – 13
v(t) = -9.0t2 + 4.0t + 4.0 (first derivative)
a(t) = -18.0t + 4.0 (second derivative)
Now we can evaluate the second derivative at t = 2.3 s:
a(2.3 s) = -18.0(2.3) + 4.0 = -36.4 m/s^2
The resultant force acting on the particle at time 2.3 s can be found using Newton's second law:
F = ma
F = (0.225 kg)(-36.4 m/s^2)
F = -8.19 N
The negative sign indicates that the force is acting in the opposite direction of the particle's motion. Therefore, the resultant force acting on the particle at time 2.3 s is 8.19 N, directed opposite to the particle's motion.
To calculate the resultant force acting on the particle at time 2.3 s, we need to find the derivative of the displacement function with respect to time, and then substitute the value of time into the derivative.
Step 1: Find the derivative of the displacement function x(t) = -3.0t^3 + 2.0t^2 + 4.0t - 13.
The derivative of x(t) is given by:
dx(t)/dt = -9.0t^2 + 4.0t + 4.0
Step 2: Substitute the value of time, t = 2.3 s, into the derivative:
dx(2.3)/dt = -9.0(2.3)^2 + 4.0(2.3) + 4.0
Step 3: Calculate the value of dx(2.3)/dt:
dx(2.3)/dt = -9.0(5.29) + 9.2 + 4.0
dx(2.3)/dt = -47.61 + 9.2 + 4.0
dx(2.3)/dt = -34.41
Step 4: Now, we can use Newton's second law of motion, which states that the resultant force on an object is equal to its mass multiplied by its acceleration.
The mass of the particle is given as 225 g, which is equal to 0.225 kg.
The acceleration of the particle can be calculated using the second derivative of the displacement function:
d^2x(t)/dt^2 = -18.0t + 4.0
Substituting the value of time, t = 2.3 s, into the second derivative:
d^2x(2.3)/dt^2 = -18.0(2.3) + 4.0
d^2x(2.3)/dt^2 = -41.4 + 4.0
d^2x(2.3)/dt^2 = -37.4
Step 5: Calculate the acceleration, a, using the formula F = ma:
F = m * a
F = 0.225 kg * (-37.4 m/s^2)
F = -8.415 N
Therefore, the resultant force acting on the particle at time 2.3 seconds is -8.415 N.
To calculate the resultant force acting on the particle at time 2.3 s, we need to find the acceleration of the particle at that time and then use Newton's second law of motion, which states that force (F) is equal to mass (m) times acceleration (a).
Firstly, we need to find the derivative of the displacement function to obtain the velocity function, and then take the derivative of the velocity function to obtain the acceleration function.
Given x(t) = -3.0t^3 + 2.0t^2 + 4.0t - 13, let's find the derivative of x(t) to find the velocity function:
v(t) = dx(t)/dt
Differentiating each term with respect to t, we get:
v(t) = -9.0t^2 + 4.0t + 4.0
Now, we can find the derivative of v(t) to find the acceleration function:
a(t) = dv(t)/dt
Differentiating each term with respect to t, we get:
a(t) = -18.0t + 4.0
Now, we have the acceleration function, a(t) = -18.0t + 4.0.
To find the resultant force, we need to calculate the acceleration at time t = 2.3 s. So, substitute t = 2.3 into the acceleration function:
a(2.3) = -18.0(2.3) + 4.0
a(2.3) = -41.4 + 4.0
a(2.3) = -37.4 m/s^2
The acceleration of the particle at time 2.3 s is approximately -37.4 m/s^2.
Now, we can use Newton's second law to find the resultant force.
F = m * a
Given that the mass (m) of the particle is 225 g = 0.225 kg, we can calculate the force:
F = 0.225 kg * -37.4 m/s^2
F = - 8.415 N
Therefore, the resultant force acting on the particle at time 2.3 s is approximately -8.415 N.