rotating block on table URGRNT HELP

Question

A small block of mass 1.07 kg slides without friction on a horizontal table. Initially it moves in a circle of radius R₀ = 0.57 m with a speed 1.4 m/s. It is held in its path by a string that passes through a small hole at the center of the circle. The string is then pulled down a distance of 0.11 m.( R₀ - R_f = 0.11 m. It is pulled so slowly that the object continues to move in a circle of continually decreasing radius.

1- What is the torque placed on the mass by the string?

2- What is the final speed of the block?

3 - How much work was done by the force that pulled on the string?

Answer 1

Since there is no external torque exerted on the system (block+ string), angular moment L=Iω=constant, I= mR²;

hence, ω_f= I_inω_in/I_fin = I_inV_in/(R_inI_fin)=3.77 rad/s

then V_f= R_finω_f=1.73 m/s

1-   What is the torque placed on the mass by the string? Since the circular motion due to centripetal force, then τ=RF_cent

Thus,  initial torque τ₀=F₀R₀=(mV²/R₀)R₀=mV²=1.07*(1.4)²≈2.1Nm τ_f=F_fR_f=(mV_f²/R_f)R_f=mV_f²≈3.22 Nm 

2- What is the final speed of the block? V_f=1.73 m/s

3 - How much work was done by the force that pulled on the string? W=∆EK=(m/2) (V_fin²  - V_in²)=0.55 J