A block rests on a horizontal, frictionless surface. A string is attached to the block, and is pulled with a force of 45.0 N at an angle è above the horizontal, as shown in Figure 7-23. After the block is pulled through a distance of 1.50 m its speed is v = 2.90 m/s, and 50.0 J of work has been done on it.
To find the mass of the block, we can use the work-energy principle which states that the work done on an object is equal to the change in its kinetic energy. The work done on the block can be calculated using the formula W = F * d * cos(θ), where W is the work done, F is the force applied, d is the displacement, and θ is the angle between the force and the displacement.
In this case, we are given that the work done on the block is 50.0 J, the force applied is 45.0 N, and the displacement is 1.50 m. The angle is not provided explicitly in the given information, so we will assume it as 0° (horizontal). Therefore, the formula can be simplified to W = F * d.
Plugging in the values, we have:
50.0 J = 45.0 N * 1.50 m
Now, we can solve for the mass of the block. The work done is equal to the change in kinetic energy, which can be calculated using the formula:
W = ΔKE = (1/2) * m * (v^2 - u^2)
Here, u is the initial velocity, which is 0 m/s since the block is initially at rest. We are given the final velocity v as 2.90 m/s.
Substituting the values in the equation, we have:
50.0 J = (1/2) * m * (2.90^2 - 0^2)
Simplifying further, we get:
50.0 J = (1/2) * m * (2.90^2)
Now, we can solve for the mass m by rearranging the equation:
m = (2 * 50.0 J) / (2.90^2)
Evaluating the expression:
m ≈ 10.617 kg (rounded to three decimal places)
Therefore, the mass of the block is approximately 10.617 kg.