A 0.40kg block is held against a compressed spring of stiffness 68.2N/m and compression 0.64m parallel to a 24 degree incline that is 1.27m long. The coefficient of kinetic friction between the block and the incline is 0.34. The external force holding the block against the compressed spring is then removed, and the compressed spring pushes the block up the incline from rest. What is the work done by gravity on the block along the length of the incline?

What is the work done by kinetic friction along the length of the incline?

What is the work done by the spring while the block is in contact with it?

What is the speed with which the block leaves the incline?

To find the work done by gravity on the block along the length of the incline, we need to calculate the component of the gravitational force acting parallel to the incline.

Work done by gravity = Force of gravity x displacement along the incline

The force of gravity acting on the block is given by:

Force of gravity = mass x acceleration due to gravity
= 0.40kg x 9.8m/s^2
= 3.92N

The component of the force of gravity acting parallel to the incline is given by:

Force parallel = Force of gravity x sin(θ)
= 3.92N x sin(24°)
= 1.57N

The displacement along the incline is given as 1.27m.

Work done by gravity = Force parallel x displacement along the incline
= 1.57N x 1.27m
= 1.9969 J (rounded)

Therefore, the work done by gravity on the block along the length of the incline is approximately 1.9969 Joules.

Now let's calculate the work done by kinetic friction along the length of the incline.

Work done by kinetic friction = Force of friction x displacement along the incline

The force of friction can be found using the equation:

Force of friction = coefficient of kinetic friction x normal force
= μ x mass x acceleration due to gravity
= 0.34 x 0.40kg x 9.8m/s^2
= 1.34N

The displacement along the incline is given as 1.27m.

Work done by kinetic friction = Force of friction x displacement along the incline
= 1.34N x 1.27m
= 1.6998 J (rounded)

Therefore, the work done by kinetic friction along the length of the incline is approximately 1.6998 Joules.

Next, let's determine the work done by the spring while the block is in contact with it.

The work done by the spring can be calculated using the equation:

Work done by the spring = (1/2) x k x x^2

Here, k represents the stiffness of the spring (68.2N/m) and x represents the compression (0.64m).

Work done by the spring = (1/2) x 68.2N/m x (0.64m)^2
= (1/2) x 68.2N/m x 0.4096m^2
= 13.965 J (rounded)

Therefore, the work done by the spring while the block is in contact with it is approximately 13.965 Joules.

Finally, let's find the speed with which the block leaves the incline.

The initial potential energy stored in the spring is equal to the work done by the spring:

Potential energy stored in the spring = Work done by the spring
= 13.965 J

This potential energy is converted to kinetic energy when the block reaches the top of the incline. Therefore:

Potential energy = Kinetic energy

(1/2) x mass x velocity^2 = 13.965 J

Considering the mass of the block as 0.40kg, we can rearrange the equation:

(1/2) x 0.40kg x velocity^2 = 13.965 J

velocity^2 = (13.965 J x 2) / 0.40kg
velocity^2 = 69.825 m^2/s^2

Taking the square root of both sides:

velocity = √69.825 m^2/s^2
velocity ≈ 8.359 m/s

Therefore, the speed with which the block leaves the incline is approximately 8.359 m/s.

To find the work done by gravity on the block along the length of the incline, we can use the formula:

Work = Force * Distance * cos(θ)

where Force is the weight of the block acting parallel to the incline, Distance is the length of the incline, and θ is the angle of the incline.

First, let's calculate the weight of the block:

Weight = mass * gravity
Weight = 0.40 kg * 9.8 m/s^2 = 3.92 N

Now, let's calculate the component of the weight parallel to the incline:

Force = Weight * sin(θ)
Force = 3.92 N * sin(24°) = 1.58 N

Since the block is moving up the incline, the angle between the force of gravity and the direction of motion is 180°, so we take the cosine of 180°, which is -1. Therefore:

Work = 1.58 N * 1.27 m * cos(180°) = -2.01 J

The negative sign indicates that the work done by gravity is in the opposite direction of motion.

Next, let's find the work done by kinetic friction along the length of the incline. The formula for the work done by friction is:

Work = Force of friction * Distance

The force of friction can be calculated using the formula:

Force of friction = coefficient of friction * normal force

The normal force is the component of the weight perpendicular to the incline:

Normal force = Weight * cos(θ)
Normal force = 3.92 N * cos(24°) = 3.52 N

Now, let's calculate the force of friction:

Force of friction = 0.34 * 3.52 N = 1.20 N

Finally, let's calculate the work done by friction:

Work = 1.20 N * 1.27 m = 1.52 J

The work done by friction is positive because it is acting in the opposite direction of motion.

To find the work done by the spring while the block is in contact with it, we use the formula:

Work = (1/2) * k * compression^2

where k is the stiffness of the spring and compression is the amount the spring is compressed.

Work = (1/2) * 68.2 N/m * (0.64 m)^2 = 13.09 J

The work done by the spring is positive because it is acting in the direction of motion.

Finally, to find the speed with which the block leaves the incline, we can use the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy:

Work done by all forces = change in kinetic energy

The work done by gravity is -2.01 J, the work done by friction is 1.52 J, and the work done by the spring is 13.09 J. The total work done is the sum of these values:

Total work = -2.01 J + 1.52 J + 13.09 J = 12.60 J

The change in kinetic energy is equal to the final kinetic energy, which is zero since the block comes to a stop, minus the initial kinetic energy, which is also zero since the block is at rest. Therefore:

Change in kinetic energy = 0 - 0 = 0 J

Setting the total work equal to the change in kinetic energy:

12.60 J = 0 J

Since the block comes to a stop, it means that all the energy from the spring is transformed into other forms of energy (such as potential energy), and there is no kinetic energy remaining. Therefore, the speed with which the block leaves the incline is 0 m/s.