The 20lb cartis released from rest at time t=0 on the incline surface. The force P=8t lb acts on the cart, where t is the time measured in seconds. A) Determine the distance the cart will move down the inclined surface before reversing direction. B) Find the velocity of the cart when it returns to the point of release.
The angle of inclination is 20 degrees
To determine the distance the cart will move down the inclined surface before reversing direction, we need to analyze the forces acting on the cart.
Let's start by finding the net force acting on the cart at any given time t. The net force is the difference between the force due to gravity and the force P.
Force due to gravity (Fg) can be calculated using the formula: Fg = m * g, where m is the mass of the cart and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given that the cart has a weight of 20 lb, we can convert it to mass by dividing by the gravitational acceleration: m = 20 lb / 9.8 m/s^2.
Now we can calculate the force due to gravity: Fg = m * g.
Next, we need to calculate the component of gravity force parallel to the inclined surface. This can be found using: Fg_parallel = Fg * sin(theta), where theta is the angle of inclination (20 degrees).
We can then calculate the net force (Fn) acting on the cart by subtracting the force P (8t lb) from the parallel component of gravity force: Fn = Fg_parallel - P.
To determine the distance the cart will move down the inclined surface before reversing direction, we need to find the time (t_rev) at which the net force (Fn) becomes zero. This will be the point where the direction of motion changes.
Setting Fn equal to zero: 0 = Fg_parallel - P.
Solving for t_rev, we get: t_rev = Fg_parallel / P.
Substituting the expressions for Fg_parallel and P, we obtain: t_rev = (m * g * sin(theta)) / (8t).
Now we can calculate the distance traveled before reversing direction. The distance (d) traveled can be found using the equation of motion for uniform acceleration: d = (1/2) * a * t_rev^2, where a is the acceleration.
To find the acceleration, we can use the second law of motion: Fn = m * a, where Fn is the net force (which is equal to zero when the cart reverses direction).
Solving for the acceleration, we get: a = Fn / m.
Substituting the expression for Fn and the known values, we obtain: a = (Fg_parallel - P) / m.
Finally, we can calculate the distance traveled down the incline before reversing direction: d = (1/2) * a * t_rev^2.
To find the velocity of the cart when it returns to the point of release, we need to calculate the final velocity (v_f) using the equation: v_f = a * t_rev.
Substituting the known values, we can find the velocity.