Two objects are connected by a massless string as shown in the figure below. The pulley is massless and rotates without friction. The object of mass m = 1.7 kg slides without friction on an inclined plane which makes an angle of è = 38° with the horizontal. The object of mass M = 3.2 kg, hangs from the string. If the two objects are released from rest with the string taut, what is their total kinetic energy when the object of mass M has fallen 26 cm?

Question

Two objects are connected by a massless string as shown in the figure below. The pulley is massless and rotates without friction. The object of mass m = 1.7 kg slides without friction on an inclined plane which makes an angle of è = 38° with the horizontal. The object of mass M = 3.2 kg, hangs from the string. If the two objects are released from rest with the string taut, what is their total kinetic energy when the object of mass M has fallen 26 cm?

Answer 1

without the figure, hard to know the situation.

Answer 2

To find the total kinetic energy when the object of mass M has fallen 26 cm, we need to calculate the final velocities of both objects and then use the kinetic energy formula.

First, let's find the final velocity of the object of mass M. We can use the equation of motion for an object in free fall:

v^2 = u^2 + 2as

Here, v is the final velocity, u is the initial velocity (which is 0 since it is released from rest), a is the acceleration due to gravity (9.8 m/s^2), and s is the distance fallen by the object (26 cm = 0.26 m).

v^2 = 0^2 + 2 * 9.8 * 0.26
v^2 = 5.096

Taking the square root of both sides, we get:

v = √5.096
v ≈ 2.26 m/s

Now, let's find the final velocity of the object of mass m. The inclined plane has an angle of θ = 38° with the horizontal, so the component of the acceleration along the incline is given by:

a_parallel = g * sin(θ)

a_parallel = 9.8 * sin(38°)
a_parallel ≈ 5.926 m/s^2

Next, we can use the equation of motion for an object sliding on an inclined plane:

v^2 = u^2 + 2as

Here, v is the final velocity, u is the initial velocity (which is 0 since it is released from rest), a is the acceleration along the incline (a_parallel), and s is the distance traveled by the object (which is the same as the distance fallen by object M, 0.26 m).

v^2 = 0^2 + 2 * 5.926 * 0.26
v^2 = 3.07

Taking the square root of both sides, we get:

v = √3.07
v ≈ 1.75 m/s

Now that we have the final velocities of both objects, we can calculate their kinetic energy.

The kinetic energy formula is:

KE = 0.5 * m * v^2

For the object of mass M:

KE_M = 0.5 * 3.2 * (2.26)^2

And for the object of mass m:

KE_m = 0.5 * 1.7 * (1.75)^2

The total kinetic energy is the sum of the individual kinetic energies:

Total KE = KE_M + KE_m

You can plug in the values and calculate the total kinetic energy.