May one who prays Amidah assist his son who doesn’t know what to say at the Tefillah and is hinting for help?
Although one isn’t allowed to hint during the Amidah just like he isn’t allowed to do so during the Shema, if one is interrupted in any way he may stop and hint as needed. Examples to that are brought down in the Mishna berura, of a baby who’s crying, that it may be hushed and hinted at to stop. Another example would be if the chazzan waits for the person praying to end his Tefilla, he may hint to him that he should start without him.
Accordingly, Rav Shtarnbuch shlita writes, if one’s child needs assistance with his prayer, the father may stop his praying without talking and show him how to continue. The reason being, the father’s kavana is disturbed by the fact that the child doesn’t know how to pray. As opposed to it being just for the sake of the child’s prayer. In that case, he shouldn’t stop. Harav Chaim Kanievsky Shlita on the other hand, said that if one’s child is confused with the Tefilla, the father may show him how to proceed without talking even if it’s solely for the sake of the child’s tefilla and it doesn’t bother the father’s Tefilla at all.
Is one allowed to ride his bicycle on Shabbat?
There are three issues concerning riding a bike. One is the prohibition riding itself might have. The second is carrying in a place without eruv. Finally, the fact that riding might not be within the spirit of Shabbat which could be called זלזול שבת.
The first concern was addressed by many poskim which some prohibited due to a concern he might ride the bike and cross the Techum Shabbat (kaf hachayim). Others felt it should be restricted since the bike might break and he might come to fix it which is a Biblical prohibition of “בונה – building”. Others just felt it might be asur because of חורש when he rides on the earth.
The Ben Ish Chai however rejected all those reasonings, saying that all the concerns raised can be disputed, since chazal didn’t restrict the riding, and after the times of the Gemara, no one has the authority to make new decrees.
The Ben Ish Chai concludes that riding a bike is permitted if the place is gated or it has a valid eruv. But he is the only one who permitted as all the other poskim were stringent.
Even those who agreed with The Ben Ish Chai’s reasoning still felt its עובדין דחול and זלזול שבת.
Similarly, after lengthy discussion, Hacham Ovadia felt one isn’t allowed to ride a bike.
May one exercise on Shabbat?
Although exercising for the sake of losing weight through sweating is unanimously prohibited, as it’s considered part of the medical decree chazal instituted. If one wants to exercise for enjoyment, it is permitted.
The reasoning being that the Shulchan aruch writes that only if one sweats it should be avoided.
The poskim argued whether it’s permitted to use exercise tools and equipment.
The Ohr Letsion wrote it’s permitted even while using equipment as lifting weights etc.
Other poskim felt that exercising with equipment isn’t permitted in any form since they’re muktse, and using them would also constitute עובדין דחול.
 סימן קד סק״א
 תשובות והנהגות ח״ג סי לה
 שיח תפלה ספר הזיכרון, הובא בדרשו
 רב פעלים חלק א’ אורח חיים סימן כה
 שו״ת חזון עובדיה שבת ח”ד עמוד מג, ושו”ת יביע אומר ח”י
 אור לציון ח״ב עמוד רנח
 סי’ שכח סעיף מב