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A ball of mass 0.35 kg is attached to the end of a horizontal cord and is rotated in a circle of radius 1.0 m on a frictionless horizontal surface. If the cord breaks when the tension in it exceeds 80 N, determine: (a) the maximum speed of the ball, (b) the minimum period of the ball.
alphonsio
  • (a) The maximum speed of the ball is approximately 15.11m/s15.11 \, \text{m/s}.

  • (b) The minimum period of the ball is approximately 0.416s0.416 \, \text{s}.


A ball of mass 0.35 kg is attached to the end of a horizontal cord and is rotated in a circle 
of radius 1.0 m

To solve this problem, we’ll first use the relationship between centripetal force, mass, velocity, and radius. The centripetal force FcF_c required to keep an object of mass mm moving in a circle of radius rr with a speed vv is given by:

Fc=mv2rF_c = \frac{mv^2}{r}

Given:

  • Mass, m=0.35kgm = 0.35 \, \text{kg}
  • Radius, r=1.0mr = 1.0 \, \text{m}
  • Maximum tension (which is the centripetal force at breaking point), Fc=80NF_c = 80 \, \text{N}

(a) Maximum Speed of the Ball

We need to determine the maximum speed vmaxv_{\text{max}} at which the cord can hold the ball without breaking. Using the centripetal force formula:

v2=r×Fcmv^2 = \frac{r \times F_c}{m}

v=r×Fcmv = \sqrt{ \frac{r \times F_c}{m}}

Solving for vmaxv_{\text{max}}:

vmax=1m×80N0.35kgv_{max} = \sqrt{ \frac{1 \, \text{m} \times 80 \, \text{N}}{0.35 \, \text{kg}}}

vmax228.57v_{\text{max}} \approx \sqrt{228.57}

vmax15.1185789204m/sv_{\text{max}} \approx 15.1185789204 \, \text{m/s}

(b) Minimum Period of the Ball

The period TT is the time it takes for the ball to make one complete revolution around the circle. The relationship between the speed vv, the period TT, and the radius rr of the circle is given by:

v=2πrTv = \frac{2\pi r}{T}

Solving for the period TT:

T=2πrvT = \frac{2\pi r}{v}

Using the maximum speed vmaxv_{\text{max}}:

Tmin=2π1.0m15.11m/sT_{\text{min}} = \frac{2\pi \cdot 1.0 \, \text{m}}{15.11 \, \text{m/s}}

Tmin2π15.11T_{\text{min}} \approx \frac{2\pi}{15.11}

Tmin6.2815.11T_{\text{min}} \approx \frac{6.28}{15.11}

Tmin0.4155936441sT_{\text{min}} \approx 0.4155936441 \, \text{s}

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