The puck in the figure below has a mass of 0.160 kg. Its original distance from the center of rotation is 40.0 cm, and it moves with a speed of 60.0 cm/s. The string is pulled downward 15.0 cm through the hole in the frictionless table. Determine the work done on the puck. (Hint: Consider the change of kinetic energy of the puck.)
To determine the work done on the puck, we need to consider the change in kinetic energy. The work done on an object is equal to the change in kinetic energy.
The initial kinetic energy of the puck is given by:
KE_initial = 0.5 * m * v_initial^2
where m is the mass of the puck (0.160 kg) and v_initial is its initial speed (60.0 cm/s).
Substituting the given values:
KE_initial = 0.5 * 0.160 kg * (60.0 cm/s)^2
Next, we need to find the final kinetic energy of the puck. Since the string is pulled downward, the puck moves in a circular path. The work done on the puck is due to the change in its kinetic energy as it moves downward:
KE_final = KE_initial + Work
The final kinetic energy can be calculated using the mass, velocity, and distance from the center of rotation:
KE_final = 0.5 * m * v_final^2
To find the final velocity, we can use the conservation of angular momentum. The initial angular momentum is given by:
L_initial = m * r * v_initial
where r is the initial distance from the center of rotation (40.0 cm). The final angular momentum is given by:
L_final = m * (r - h) * v_final
where h is the distance the string is pulled downward (15.0 cm).
Since angular momentum is conserved, we can equate the initial and final angular momentum:
L_initial = L_final
m * r * v_initial = m * (r - h) * v_final
Simplifying the equation gives us:
v_final = (r * v_initial) / (r - h)
Substituting the given values:
v_final = (40.0 cm * 60.0 cm/s) / (40.0 cm - 15.0 cm)
Now, we can calculate the final kinetic energy:
KE_final = 0.5 * m * v_final^2
Substituting the given values:
KE_final = 0.5 * 0.160 kg * [(40.0 cm * 60.0 cm/s) / (40.0 cm - 15.0 cm)]^2
Finally, we can calculate the work done on the puck:
Work = KE_final - KE_initial
Substituting the calculated values of KE_final and KE_initial:
Work = [0.5 * 0.160 kg * [(40.0 cm * 60.0 cm/s) / (40.0 cm - 15.0 cm)]^2] - [0.5 * 0.160 kg * (60.0 cm/s)^2]
To determine the work done on the puck, we need to consider the change in kinetic energy. The work-energy principle states that the work done on an object is equal to the change in its kinetic energy.
The initial kinetic energy of the puck is given by K1 = (1/2)mv1^2, where m is the mass of the puck (0.160 kg) and v1 is its initial speed (60.0 cm/s).
The final kinetic energy of the puck is given by K2 = (1/2)mv2^2, where v2 is the final speed of the puck. To find v2, we can use the conservation of angular momentum.
Angular momentum is given by L = Iω, where I is the moment of inertia and ω is the angular velocity. In this case, the moment of inertia is equal to m*r^2, where r is the distance from the center of rotation (40.0 cm). The angular velocity ω is related to the linear velocity v2 by the equation ω = v2/r.
We can substitute the expressions for I and ω into the equation for angular momentum to get mv1r = m*r^2*(v2/r), which simplifies to v1 = r*v2.
From this equation, we can solve for v2: v2 = v1/r.
Substituting this value of v2 into the expression for K2, we get K2 = (1/2)m(v1/r)^2.
Therefore, the work done on the puck is given by W = K2 - K1 = (1/2)m(v1/r)^2 - (1/2)mv1^2.
Plugging in the given values m = 0.160 kg, v1 = 60.0 cm/s, and r = 40.0 cm, we can calculate the work done on the puck.
W=KE₂-KE₁=mv₂²/2 -mv₁²/2=
=0.5m(v₂² - v₁²)
Law of conservation of angular momentum
I₁ω₁=I₂ω₂
(m₁ r₁²)(v₁/r₁)=(m₂ r₂²)(v₂/r₂)
v₂ =(r₁/r₂)v₁
W= 0.5mv₁²{(r₁/r₂)²-1}