A small block on a frictionless horizontal surface has a mass of 0.0215 kg. It is attached to a massless cord passing through a hole in the surface. (See the figure below (Figure 1).) The block is originally revolving at a distance of 0.280 m from the hole with an angular speed of 1.85 rad/s. The cord is then pulled from below, shortening the radius of the circle in which the block revolves to 0.200 m. You may treat the block as a particle.
A.) Is angular momentum conserved?
B.) How do you know? Match the words in the left column to the appropriate blanks in the sentence on the right. Make certain the sentence is complete before submitting your answer.
C.) What is the new angular speed?
Express your answer in radians per second.
D.) Find the change in kinetic energy of the block.
Express your answer in joules.
E.) How much work was done in pulling the cord?
Express your answer in joules.
I need help solving this problem. I would want a step-by-step explanation so that in the future, I would be able to understand it.
A) Yes
B) The angular momentum is conserved because the torque by the acting
force is 0 as r=0.
C) mvr = conserved
So find the new v and divide by new r to get the answer
D) Change in K.E = (1/2)m(v.v-u.u)
E) Work Done=Change in K.E
Thanks for the solution!
I only need the answers for D and E.
A.) Is angular momentum conserved?
To determine if angular momentum is conserved, we need to compare the initial angular momentum to the final angular momentum. Angular momentum is conserved when there is no external torque acting on the system.
B.) How do you know? Match the words in the left column to the appropriate blanks in the sentence on the right. Make certain the sentence is complete before submitting your answer.
1. Initial angular momentum a. stays the same
2. Final angular momentum b. decreases
3. External torque c. the same
4. Conservation of angular momentum d. acts on the system
The correct matching would be:
1. Initial angular momentum c. the same
2. Final angular momentum b. decreases
3. External torque d. acts on the system
4. Conservation of angular momentum a. stays the same
Based on the above matching, if the initial and final angular momentums are the same, then angular momentum is conserved.
C.) What is the new angular speed?
To find the new angular speed, we can use the conservation of angular momentum.
The initial angular momentum is given by:
L_initial = m * r_initial * v_initial
where
m = mass of the block = 0.0215 kg
r_initial = initial radius = 0.280 m
v_initial = initial tangential velocity = r_initial * angular speed = 0.280 m * 1.85 rad/s
The final angular momentum is given by:
L_final = m * r_final * v_final
where
r_final = final radius = 0.200 m
Since angular momentum is conserved, L_initial = L_final.
m * r_initial * v_initial = m * r_final * v_final
Simplifying, we can solve for v_final:
r_initial * v_initial = r_final * v_final
v_final = (r_initial * v_initial) / r_final
Plug in the given values and calculate v_final.
D.) Find the change in kinetic energy of the block.
To find the change in kinetic energy, we can use the formula:
ΔKE = KE_final - KE_initial
where
KE_initial = 1/2 * m * (v_initial)^2
KE_final = 1/2 * m * (v_final)^2
Plug in the initial and final velocities, and calculate ΔKE.
E.) How much work was done in pulling the cord?
To find the work done, we can use the work-energy principle.
The work done on an object is equal to the change in its kinetic energy. Therefore, the work done in pulling the cord is the same as the change in kinetic energy calculated in question D.
To solve this problem, we can use the principle of conservation of angular momentum. Here's a step-by-step explanation of how to solve this problem:
A.) To determine if angular momentum is conserved, we need to compare the initial and final angular momentum of the system. Angular momentum is defined as the product of the moment of inertia and the angular velocity.
Angular momentum (L) = moment of inertia (I) * angular velocity (ω)
B.) Let's list down the known values for the initial and final conditions:
Initial:
Mass of the block (m) = 0.0215 kg
Initial radius (r) = 0.280 m
Initial angular speed (ω) = 1.85 rad/s
Final:
Final radius (r) = 0.200 m
Unknown final angular speed (ω_f)
The statement "We may treat the block as a particle" implies that the mass distribution of the block is concentrated at a single point, simplifying the calculation of moment of inertia.
C.) To find the new angular speed, we can use the concept of conservation of angular momentum:
Initial angular momentum (L_i) = Final angular momentum (L_f)
Initial moment of inertia (I_i) * Initial angular velocity (ω_i) = Final moment of inertia (I_f) * Final angular velocity (ω_f)
The moment of inertia of a point mass rotating about an axis at a distance (r) is given by:
I = m * r^2
Let's calculate the initial and final moment of inertia:
Initial moment of inertia (I_i) = m * r_i^2
Final moment of inertia (I_f) = m * r_f^2
Substituting the given values:
I_i = (0.0215 kg) * (0.280 m)^2
I_f = (0.0215 kg) * (0.200 m)^2
Now, using the conservation of angular momentum equation:
(I_i * ω_i) = (I_f * ω_f)
Substituting the values:
(0.0215 kg) * (0.280 m)^2 * 1.85 rad/s = (0.0215 kg) * (0.200 m)^2 * ω_f
Simplifying the equation, we can solve for ω_f:
(0.280^2 * 1.85) / (0.200^2) = ω_f
Calculating the value of ω_f will give us the final angular speed.
D.) To find the change in kinetic energy of the block, we need to calculate the initial and final kinetic energies.
Initial kinetic energy (KE_i) = (1/2) * m * (r_i)^2 * (ω_i)^2
Final kinetic energy (KE_f) = (1/2) * m * (r_f)^2 * (ω_f)^2
Substituting the given values:
KE_i = (1/2) * (0.0215 kg) * (0.280 m)^2 * (1.85 rad/s)^2
KE_f = (1/2) * (0.0215 kg) * (0.200 m)^2 * (ω_f)^2
Calculating the values of KE_i and KE_f, we can find the change in kinetic energy (ΔKE) by subtracting the initial kinetic energy from the final kinetic energy:
ΔKE = KE_f - KE_i
E.) To calculate the work done in pulling the cord, we need to use the work-energy principle. The work done (W) is equal to the change in kinetic energy.
Work done (W) = ΔKE
Substituting the value of ΔKE calculated previously will give us the work done.
By following these steps, you should be able to solve the problem and find the answers to all the questions (A, B, C, D, and E).