A car starts rolling down a l-in-4 hill (l-in-4 means that for each 4 m traveled along the road, the elevation change is 1 m). How fast is it going when it reaches the bottom after traveling 55 m? (a) Ignore friction, (b) Assume an effective coefficient of friction equal to 0.10.
and also
Two boxes, mx — 1.0 kg with a coefficient of kinetic fric tion of 0.10, and m2 = 2.0 kg with a coefficient of 0.20, are placed on a plane inclined at 9 = 30°. (a) What acceleration does each box experience? (b) If a taut string is connected to the boxes (Fig. 4-64), with m2 initially farther down the slope, what is the acceleration of each box? (c) If the initial configuration is reversed with m\ starting lower with a taut string, what is the acceleration of each box?
first:
the normal component of weight is mg*sqrt(15/16)
friction then is mg*sqrt(15/16)*mu
So going 55m, work is mg*sqrt(15/16)*.1*55
final KE=initialPE-workdone.
second question
I dontknow the incline.
the incline is 30 degrees
To calculate the speed of the car when it reaches the bottom of the hill after traveling 55 m, we need to consider the gravitational potential energy and the work done by friction (assuming friction is present).
(a) Ignoring friction:
When ignoring friction, the only force acting on the car is gravity. The potential energy loss due to the change in elevation is converted into kinetic energy.
Step 1: Calculate the change in potential energy:
The vertical distance traveled can be calculated as (55 m) / (4 m) = 13.75 m.
The change in potential energy can be calculated as:
Potential energy change = mass * gravity * change in height
= mass * g * height
= mass * 9.8 m/s^2 * 13.75 m
Step 2: Convert the potential energy change into kinetic energy:
The kinetic energy at the bottom of the hill is equal to the potential energy lost, since there is no other work done.
At the bottom, the potential energy is zero, so the kinetic energy is equal to the potential energy change:
Kinetic energy = Potential energy change
Step 3: Calculate the speed using the kinetic energy:
Kinetic energy = 1/2 * mass * velocity^2
Set the potential energy change equal to the kinetic energy and solve for velocity:
mass * 9.8 m/s^2 * 13.75 m = 1/2 * mass * velocity^2
Simplifying the equation, we find:
velocity^2 = 2 * 9.8 m/s^2 * 13.75 m
velocity = sqrt(2 * 9.8 m/s^2 * 13.75 m)
(b) Assuming a coefficient of friction equal to 0.10:
When considering friction, we need to take into account the work done against it. The work done by friction is given by the product of the force of friction and the distance.
Step 1: Calculate the work done against friction:
The vertical distance traveled is still 13.75 m.
The horizontal distance traveled can be calculated as (4 m) * (13.75 m) = 55 m.
The work done by friction is given by:
Frictional force = coefficient of friction * normal force
The normal force is equal to the weight of the car, which is mass * gravity.
The work done against friction is given by:
Work = frictional force * distance = coefficient of friction * normal force * distance
Step 2: Calculate the change in potential energy taking friction into account:
The potential energy change is still the same as in part (a) - mass * 9.8 m/s^2 * 13.75 m.
Step 3: Convert the potential energy change into kinetic energy:
The kinetic energy at the bottom of the hill is equal to the sum of the potential energy lost and the work done against friction:
Kinetic energy = Potential energy change + Work
Step 4: Calculate the speed using the kinetic energy:
Set the kinetic energy equal to 1/2 * mass * velocity^2 and solve for velocity.
Now, let's move on to the second question about the boxes on an inclined plane:
(a) Acceleration experienced by each box:
The force acting parallel to the incline is given by the weight of each box multiplied by the sine of the angle of the incline.
For m1:
Force parallel to the incline = m1 * g * sin(30°)
Acceleration = Force parallel to the incline / m1
For m2:
Force parallel to the incline = m2 * g * sin(30°)
Acceleration = Force parallel to the incline / m2
(b) Acceleration of each box when connected by a taut string:
When the two boxes are connected by a taut string, they will experience the same acceleration. This is because the tension in the string will cause both boxes to move together.
(c) Acceleration of each box when m1 starts lower with the taut string:
The acceleration calculation remains the same as in part (b). The initial configuration of the boxes does not affect their acceleration.
To answer the first question about the car rolling down the hill, we can use the principles of conservation of energy.
(a) Assuming no friction, the only force acting on the car is gravity. The potential energy lost as the car moves down the hill is converted into kinetic energy. The formula for the potential energy of an object at a certain height is given by:
Potential Energy = mass * gravity * height
In this case, the height is given by the ratio of 1m for every 4m traveled. So when the car travels 55m, the height it loses is 55m/4 = 13.75m. Assuming the mass of the car is known, you can calculate the potential energy it loses.
Next, we equate the potential energy lost to the kinetic energy gained at the bottom of the hill using the formula:
Kinetic Energy = 0.5 * mass * velocity^2
By solving these equations simultaneously, you can find the velocity of the car when it reaches the bottom.
(b) Assuming there is friction, we need to take it into account. The work done by the frictional force will subtract from the potential energy and convert it into work done against friction. We can still use conservation of energy but now with this additional term. The formula becomes:
Potential Energy - Work against friction = Kinetic Energy
The work done against friction can be calculated using the formula: Work = force * distance * cos(angle), where the force is the normal force multiplied by the coefficient of friction.
Again, by solving the resulting equation(s) simultaneously, you can find the velocity of the car when it reaches the bottom.
Moving on to the second question about the boxes on an inclined plane:
(a) To find the acceleration of each box, we need to consider the forces acting on them. The force of gravity can be split into components parallel and perpendicular to the inclined plane. The force parallel to the plane is responsible for acceleration, while the force perpendicular to the plane is balanced by the normal force. The force parallel to the plane is given by:
Force_parallel = mass * gravity * sin(angle)
Then, we can use Newton's second law, which states that Force = mass * acceleration, to find the accelerations of each box.
(b) If a taut string is connected to the boxes, they will experience the same acceleration because they are connected. The tension in the string will provide an additional force that affects both boxes equally.
(c) If the initial configuration is reversed, with m2 starting lower, the force of gravity acting on m2 will be stronger. Therefore, m2 will accelerate faster than m1, assuming the force of friction does not prevent it from doing so.