If motor M exerts a force of F=(10t^2+100)N on the cable, where t is in seconds, determine the velocity of the 25kg crate when t=4s. The coefficients of static and kinetic friction between the crate and the place are ÃŽ¼s=0.3 and ÃŽ¼k = 0.25, respectively. The crate is intially at rest.
To determine the velocity of the 25kg crate when t = 4s, we need to consider the forces acting on the crate and apply Newton's second law of motion.
First, let's break down the forces acting on the crate:
1. The force of the motor M: F = (10t^2 + 100) N
2. The weight of the crate: mg, where m = 25kg and g = 9.8 m/s^2 (acceleration due to gravity)
3. The force of friction: F_friction
Since the crate is initially at rest, the force of the motor M and the force of friction will be opposing forces. The crate will start moving when the force of the motor M exceeds the force of friction. This happens when the force of the motor M minus the force of friction is greater than zero, i.e., F - F_friction > 0.
Let's calculate the force of friction:
The maximum force of static friction, F_friction_max, is given by the equation F_friction_max = μs * N, where μs is the coefficient of static friction and N is the normal force. The normal force is equal to the weight of the crate, which is mg.
So, F_friction_max = μs * mg = 0.3 * 25kg * 9.8 m/s^2 = 73.5N.
Since the force of the motor M is changing with time, we need to find F_friction when t = 4s:
F = (10t^2 + 100) N
F_friction = F_friction_max = 73.5N (when the crate is on the verge of moving)
Now, let's check if the force of the motor M is greater than the force of friction:
F - F_friction > 0
(10t^2 + 100) - 73.5 > 0
10t^2 + 26.5 > 0
We can see that 10t^2 + 26.5 is always greater than zero for any value of t, which means that the force of the motor M is always greater than the force of friction. So, the crate starts moving immediately when the motor is turned on.
To find the velocity of the crate at t = 4s, we can use the equation of motion:
v = u + at,
where v is the final velocity, u is the initial velocity (which is 0 m/s since the crate is initially at rest), a is the acceleration, and t is the time.
The net force acting on the crate is the force of the motor M minus the force of friction:
net force = F - F_friction.
Using Newton's second law, F = ma, we have:
ma = F - F_friction,
where m is the mass of the crate, a is the acceleration, F is the force of the motor M, and F_friction is the force of friction.
To find the acceleration, we rearrange the equation:
a = (F - F_friction) / m.
Substituting the given values:
a = ((10t^2 + 100) - 73.5) / 25.
Now, we can calculate the acceleration at t = 4s:
a = ((10 * 4^2 + 100) - 73.5) / 25.
a = (160 + 100 - 73.5) / 25.
a = 186.5 / 25.
a = 7.46 m/s^2.
Finally, we can substitute the values of u = 0 m/s, a = 7.46 m/s^2, and t = 4s into the equation v = u + at:
v = 0 + (7.46 * 4).
v = 29.84 m/s.
Therefore, when t = 4s, the velocity of the 25kg crate is 29.84 m/s.