Question

1. For a stationary ball of mass m = 0.200 kg hanging from a massless string,...

1. For a stationary ball of mass m = 0.200 kg hanging from a massless string, draw arrows (click on the “Shapes” tab) showing the forces acting on the ball (lengths can be arbitrary, but get the relative lengths of each force roughly correct). For this case of zero acceleration, use Newton’s 2nd law to find the magnitude of the tension force in the string, in units of Newtons. Since we will be considering motion in the horizontal xy plane, we will take the z axis to be the vertical direction. The weight w = mg is in the downward z direction, while the tension T is in the upward direction of the string. The vector sum of these forces should be set equal to the mass times the acceleration of the ball, but that is zero in this case.

2. Horizontal circular motion The ball is now set into circular motion with a constant velocity v = 2.00 m/s and radius r = 0.300 m in the horizontal xy plane. Draw arrows showing the forces acting on the ball, and the direction of the centripetal acceleration. Compute the magnitude of the acceleration, and also the period Tp of the motion (the time for one complete revolution around the circle).

3. To find the angle φ corresponding to the given velocity and radius we need to consider the x and z components of the forces in question 2, separately. Since there is no motion in the z direction, the acceleration az = 0. Hence the sum of the z components of the forces must be equal to zero. From this, solve for an equation relating the tension force in the string to the angle φ.

4. Take the x axis to be centered on the ball, and pointing radially inward toward the center of the circle. Newton’s 2nd law now states that the sum of the x components of the forces in question 2 must be equal to the mass times the centripetal acceleration. Solve this equation for the tension force in the string, and since this must be equal to the equation you found in slide 4, find an equation relating the angle φ to the velocity and radius. Calculate φ for the values given on question 2 , and then from the equation for
T on question 2 find the tension in the string for the rotating ball .

5. Vertical circular motion The ball is now set into circular motion in an xz plane perpendicular to the ground. In the drawing below, draw and label arrows showing the forces acting on the ball at the four different positions around the circle. Also show the net acceleration direction at each position (adding the vectors for the gravitational acceleration and the centripetal acceleration). The ball will slow down going up, and speed up coming down, shown by the velocity arrows in the drawing.

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