11.8: Problem
- Page ID
- 32996
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- Show using the component form of the cross product given by equation (11.4) that A × B = -B × A.
- A mass M is sliding on a frictionless table, but is attached to a string which passes through a hole in the center of the table as shown in figure 11.8. The string is gradually drawn in so the mass traces out a spiral pattern as shown in figure 11.8. The initial distance of the mass from the hole in the table is R and its initial tangential velocity is v. After the string is drawn in, the mass is a distance R′ from the hole and its tangential velocity is v′.
- Given R, v, and R′, find v′.
- Compute the change in the kinetic energy of the mass in going from radius R to radius R′.
- If the above change is non-zero, determine where the extra energy came from. Figure 11.8: Trajectory of a mass on a frictionless table attached to a string which passes through a hole in the table. The string is drawing the mass in.
- A car of mass 1000 kg is heading north on a road at 30 m s-1 which passes 2 km east of the center of town.
- Compute the angular momentum of the car about the center of town when the car is directly east of the town.
- Compute the angular momentum of the car about the center of town when it is 3 km north of the above point.
- The apparatus illustrated in figure 11.9 is used to raise a bucket of mass M out of a well.
- What force F must be exerted to keep the bucket from falling back into the well?
- If the bucket is slowly raised a distance d, what work is done on the bucket by the rope attached to it?
- What work is done by the force F on the handle in the above case? Figure 11.9: A crank on a fixed axle turns a drum, thus winding the rope around the drum and raising the mass.
- Derive equations below.
- \(K_{\text {total }}=K_{\text {trans }}+K_{\text {intern }}=\left[M_{\text {total }} V_{c m}^{2} / 2\right]+\left[M_{1} v_{1}^{\prime 2} / 2+M_{2} v_{2}^{\prime 2} / 2\right] \text { . }\)
- \(\mathbf{L}_{\text {total }}=\mathbf{L}_{\text {orb }}+\mathbf{L}_{\text {spin }}=\left[M_{\text {total }} \mathbf{R}_{c m} \times \mathbf{V}_{c m}\right]+\left[M_{1} \mathbf{r}_{1}^{\prime} \times \mathbf{v}_{1}^{\prime}+M_{2} \mathbf{r}_{2}^{\prime} \times \mathbf{v}_{2}^{\prime}\right]\)
- A mass M is held up by the structure shown in figure 11.10. The support beam has negligible mass. Find the tension T in the diagonal wire. Hint: Compute the net torque on the support beam about point A due to the tension T and the weight of the mass M. Figure 11.10: A mass is supported by the tension in the diagonal wire. The support beam is free to pivot at point A.
- A system consists of two stars, one of mass M moving with velocity v1 = (0,v, 0) at position r1 = (d, 0, 0), the other of mass 2M with zero velocity at the origin.
- Find the center of mass position and velocity of the system of two stars.
- Find the spin angular momentum of the system.
- Find the internal kinetic energy of the system. Figure 11.11: A mass is supported by two strings. Figure 11.12: A ladder leaning against a wall is held in place the force F acting on the base of the ladder.
- A solid disk is rolling down a ramp tilted an angle θ from the horizontal. Compute the acceleration of the disk down the ramp and compare it with the acceleration of a block sliding down the ramp without friction.
- A mass M is suspended from the ceiling by two strings as shown in figure 11.11. Find the tensions in the strings.
- A man of mass M is a distance D up a ladder of length L which makes an angle \(\theta\) with respect to the vertical wall as shown in figure 11.12. Take the mass of the ladder to be negligible. Find the force F needed to keep the ladder from sliding if the wall and floor are frictionless and therefore can only exert normal forces A and B on the ladder.