a mass of 60kg slides on a table. the frictional force is 60 times the velocity and the mass is being pushed with a force of 54sin2tN.find the velocity as a function of time if (v)=0 at (t)=0
To find the velocity of the mass as a function of time, we need to analyze the forces acting on it and apply Newton's second law of motion.
Let's break down the forces acting on the mass:
1. Gravity: The force exerted due to the mass of the object can be calculated using the formula F_gravity = m * g, where m is the mass of the object (60 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_gravity = (60 kg) * (9.8 m/s^2) = 588 N
2. Friction: The frictional force is given as 60 times the velocity. We can represent this as F_friction = -60 * v, where v is the velocity of the mass.
3. External force: The mass is being pushed with a force of 54sin(2t) N. This force is applied in a specific direction, which we need to consider when calculating the velocity.
Taking all these forces into account, we can apply Newton's second law of motion: F_net = m * a, where F_net is the net force, m is the mass, and a is the acceleration.
F_net = 54sin(2t) N - 60v N = m * a
Now, since we know a = dv/dt (the derivative of velocity with respect to time), we can rewrite the equation:
54sin(2t) - 60v = 60 * (dv/dt)
Simplifying further,
dv/dt + 6v = 9sin(2t)
This is a first-order linear differential equation. We can solve it using an integrating factor:
1. Multiply the entire equation by the integrating factor, which is e^(∫6dt) = e^(6t).
e^(6t) * dv/dt + 6e^(6t) * v = 9e^(6t) * sin(2t)
2. Notice that d(e^(6t) * v)/dt = e^(6t) * dv/dt + 6e^(6t) * v. Rewrite the equation using this notation:
d(e^(6t) * v)/dt = 9e^(6t) * sin(2t)
3. Integrate both sides with respect to t:
∫d(e^(6t) * v)/dt dt = ∫9e^(6t) * sin(2t) dt
e^(6t) * v = ∫9e^(6t) * sin(2t) dt
4. Evaluate the integral on the right side:
e^(6t) * v = -4.5 ∫e^(6t)(cos(2t) - cos(4t)) dt
Now, the integral on the right side is a little more complicated and involves integration by parts.
5. After evaluating the integral, you will have an expression that depends on t. Divide both sides of the equation by e^(6t) to isolate v:
v = (-4.5 / e^(6t)) * [integral expression]
This is the velocity (v) as a function of time (t) for the given scenario. The exact expression for the integral will depend on the bounds of integration or other initial conditions mentioned in the problem statement.